CALT-68-1601DOE RESEARCH ANDDEVELOPMENT REPORT

YCTP-P21-89

Local Discrete Symmetry and Quantum–Mechanical Hair

John Preskill?†

California Institute of Technology, Pasadena, CA 91125

and

Lawrence M. Krauss‡ §

Center for Theoretical Physics and Department of Astronomy,Sloane Lab, Yale University, New Haven, CT 06517

Abstract

A charge operator is constructed for a quantum field theory with an abeliandiscrete gauge symmetry, and a non–local order parameter is formulated that specifieshow the gauge symmetry is realized. If the discrete gauge symmetry is manifest, thenthe charge inside a large region can be detected at the boundary of the region, even ina theory with no massless gauge fields. This long–range effect has no classical analog;it implies that a black hole can in principle carry “quantum–mechanical hair.” If thegauge group is nonabelian, then a charged particle can transfer charge to a loop ofcosmic string via the nonabelian Aharonov–Bohm effect. The string loop can carrycharge even though there is no localized source of charge anywhere on the string orin its vicinity. The “total charge” in a closed universe must vanish, but, if the gaugegroup is nonabelian and the universe is not simply connected, then the “total charge”is not necessarily the same as the sum of all point charges contained in the universe.

? This work supported in part by the U.S. Department of Energy under Contract No. DE-AC0381-ER40050

† NSF Presidential Young Investigator‡ Research supported in part by U.S. Department of Energy§ NSF Presidential Young Investigator

1. Introduction

It was recently pointed out in Ref. [1] that local gauge invariance may have

interesting low–energy consequences even in the absence of light gauge fields. Imagine,

for example, a gauge theory with continuous gauge group G that undergoes the Higgs

mechanism at a symmetry–breaking mass scale v. If the only surviving manifest gauge

symmetry is a discrete, but nontrivial, subgroup H of G, then all gauge fields acquire

masses of order ev, where e is the gauge coupling. The effective field theory that

describes physics at energies well below the mass scale ev is a theory without gauge

fields, but which respects the local H symmetry. Particles in this effective field theory

may carry nontrivial H charges. Even though the classical electric field of a charged

particle is screened by the Higgs condensate, the charge induces quantum–mechanical

effects that can be detected in principle at arbitrarily long distances. In particular,

an H charge exhibits a nontrivial Aharonov–Bohm effect upon circumnavigating a

cosmic string of the underlying G gauge theory [2,3].

In fact, the notion of a local discreteH symmetry can be formulated even without

appealing to an underlying theory with a continuous gauge group G that contains H

[1,4]. The significance of local H symmetry can be appreciated directly if we consider

a spacetime manifold M that is not simply connected. The fields of an H gauge

theory defined on M need not be strictly periodic on the noncontractible closed loops

of M ; instead, the fields need only be periodic up to the action of an element of H.

Such boundary conditions distinguish local H symmetry from global H symmetry,

and allow the local symmetry to manifest itself through nontrivial Aharonov–Bohm

phenomena.

While these observations are elementary, their consequences are potentially pro-

found. For example, it is well–known that among the quantum numbers that charac-

terize a black hole are charges, like electric charge, that couple to massless gauge fields;

the long–range field of a charged particle survives even as the particle crosses the hori-

zon of the black hole. The possibility of discrete local symmetry suggests [1,4] that

a black hole may also be endowed with other varieties of “hair” that, while invisible

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classically, can be detected through quantum interference effects. Such quantum–

mechanical hair blurs the distinction between black holes and elementary particles,

thus encouraging the speculation that this distinction may eventually fade away en-

tirely.

Furthermore, the suggestion that discrete symmetries may be gauge symmetries

provides a rationale for discrete symmetries in low–energy physics. Since no funda-

mental principle prevents global symmetries from being badly broken, one is natu-

rally reluctant to impose any global symmetries, whether continuous or discrete, on a

model field theory that purports to describe Nature. But a local symmetry, whether

continuous or discrete, is intrinsically exact. Thus, the gauge principle may justify

the discrete symmetries in various extensions of the standard model. For example,

discrete symmetries are often invoked to ensure the phenomenological viability of

models of low–energy supersymmetry. Such symmetries, if global, are jeopardized by

the wormhole fluctuations in the topology of spacetime that may occur in quantum

gravity [5,6]. But, if local, these symmetries are invulnerable to wormhole dynamics

[1].

Although the arguments contained there are persuasive, the discussion in Ref. [1]

did not provide a clearly stated definition of the charge in a theory with discrete local

symmetry, or any precise mathematical criterion for whether the charge is detectable.

In this paper we will fill these gaps in the previous discussion. We hope to clarify the

concept of discrete local symmetry, and to further explore some of its consequences.

The remainder of this paper is organized as follows: In section 2, we review the

notion of a superselection rule, and observe that the central claim of Ref. [1] is the

existence of nontrivial superselection sectors labeled by the discrete gauge charge.

We construct an operator realization of this charge in section 3 that is applicable

when the discrete gauge symmetry is abelian, and we use this charge operator in

section 4 to formulate a nonlocal order parameter that specifies how the discrete

gauge symmetry is realized. The properties of the charge operator and the order

parameter are illustrated in section 5 in the context of an explicit example — a Z2

2

lattice gauge theory coupled to a Z2 spin system. We describe in section 6 how

a continuum field theory with local discrete symmetry can be formulated without

introducing any gauge fields.

In section 7, we consider the case of nonabelian discrete gauge symmetry. We

note a topological obstruction that prevents implementation of a discrete global gauge

transformation when cosmic strings are present. One consequence of this obstruction

is the nonabelian Aharonov–Bohm effect, whereby a charged particle can exchange

charge with a loop of cosmic string. The string loop can carry charge even though

there is no localized source of charge anywhere on the string or in its vicinity. This

phenomenon can also occur if the gauge group is continuous; for example, a string

loop can carry electric charge Q if the matching condition imposed by the string does

not commute with Q. Similarly, in two spatial dimensions, charge can be carried by a

pair of vortices. We observe in section 8 that, in the case of an abelian local discrete

symmetry, the total charge contained in a closed universe must vanish. We also note

that, even if cosmic strings are absent, a nonabelian Aharonov–Bohm effect can occur

in a closed universe that is not simply connected; in this case, a charged particle can

exchange charge with a “handle” that is attached to the universe. We consider the

implications of this process in connection with wormhole physics. Section 9 contains

our conclusions.

2. Superselection Rules

In any discussion of charges in a gauge theory, the concept of a superselection

rule plays a central role. We will now review this concept, emphasizing especially the

fate of the superselection rule when a local symmetry undergoes the Higgs mechanism

[7,8].

A quantum field theory is said to respect a superselection rule if the physical

Hilbert space H can be decomposed as a direct sum of distinct sectors

H = ⊕nHn , (2.1)

3

such that every local observable O preserves the decomposition,

〈m|O|n〉 = 0 , m 6= n . (2.2)

A familiar example is quantum electrodynamics in the Coulomb phase, which has

superselection sectors labeled by the electric charge Q. The local observables of QED

are gauge–invariant operators smeared in a compact region of spacetime, and local

gauge–invariant operators always have charge Q = 0. More physically, a charged

particle is accompanied by a long–range electric field; the particle cannot be annihi-

lated unless its long–range field is annihilated as well, and no local observable can

accomplish this task. (As in any discussion of superselection rules, we assume that

the spatial volume is infinite.)

The superselection sectors of QED may also be characterized by their transforma-

tion properties under global gauge transformations. If U(ω) is the unitary operator

acting on H that represents the global gauge transformation eiω ∈ U(1), then a state

of charge Q transforms according to

U(ω) |Q〉 = eiωQ |Q〉 . (2.3)

Thus, states of definite charge are preserved by the global gauge transformations, as

a state is identified with a ray in Hilbert space. A linear combination of two states

from different charge sectors is not preserved, because U(ω) rotates the relative phase

of the two states. However, because of the superselection rule, this relative phase is

completely unobservable. It is therefore legitimate to confine our attention to a single

sector, and regard all physical states as invariant under global gauge transformations;

the property that physical states are invariant distinguishes global gauge symmetries

from ordinary global symmetries. Even though local observables preserve each charge

sector, we include all of the charge sectors in H, since an isolated object can carry

any amount of charge.

Another familiar example of a superselection rule arises if a global symmetry is

spontaneously broken. In this case, there are degenerate vacua that transform one

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into another under the action of the global symmetry. Suppose, for example, that the

global symmetry is G = U(1), that there is an order parameter φ(x) transforming

according to

U(ω) φ(x) U(ω)−1 = eiω φ(x) , (2.4)

and that there are degenerate vacua labeled by α ∈ [0, 2π) such that

〈α|φ(x)|α〉 = v eiα . (2.5)

Then a distinct superselection sector can be constructed on each α vacuum; the α–

sector is spanned by smeared polynomials in local fields acting on the α–vacuum.

Physically, the sector can change only if the phase of φ is rotated throughout an

infinite spatial volume, and no local observable can accomplish this task. Because

of the superselection rule, the degeneracy associated with a spontaneously broken

symmetry cannot be directly detected by any local observer.?

Let us return now to the case of a local symmetry, and consider the fate of the

charge superselection rule of QED when the theory is in the Higgs phase [7,8]. Suppose

that the Higgs realization of the U(1) gauge symmetry is driven by the condensation of

a scalar field φ that carries charge 1 (in units of e); that is, in the unitary gauge, φ has

a nonvanishing vacuum expectation value. If all charged fields carry charge 1, then no

nontrivial superselection rule is expected to survive in the Higgs phase. For example,

let ψ be another charge–1 scalar field. Then a ψ excitation can be annihilated by the

gauge–invariant operator φ†ψ, which in unitary gauge becomes

φ† ψ = vψ+ . . . . (2.6)

In the Higgs phase, since there is just one superselection sector, all states must trans-

form trivially under global gauge transformations; in other words, all states have

? Loosely speaking, the α–vacuum is degenerate with a zero–momentum Goldstone boson in theα–sector. Technically, though, the ground state of the α–sector is unique, since a momentumeigenstate is not normalizable.

5

charge Q = 0. Physically, the electric charge of an arbitrary state is completely

screened by the Higgs condensate. Charged particles no longer induce any effect that

can be detected at spatial infinity. (One might prefer to say that the charge Q is ill–

defined because of copious vacuum charge fluctuations. We find it more instructive

to say that Q vanishes, although defining Q is a delicate matter. We elaborate on

this viewpoint in sections 3 and 4.)

More interesting than the case in which the gauge symmetry is completely broken

is the case discussed in Ref. [1], in which a discrete subgroup of the gauge group

remains manifest. Consider now, for example, a U(1) gauge theory that contains a

charge–N field of η and a charge–1 field φ. Then, if η condenses but φ does not, there

is a surviving ZN gauge symmetry under which φ transforms as

U

(

2πk

N

)

φ U

(

2πk

N

)−1

= e2πik/N φ ,

k = 0, 1, . . . , N − 1 . (2.7)

In this theory, charge screening is incomplete, because, as we will describe in more

detail in section 3, the charge modulo N of a state is not screened by the Higgs

condensate. Thus, there is a nontrivial superselection rule. The Hilbert space de-

composes into N sectors labeled by the charge Q mod N , with states of charge Q

transforming under ZN gauge transformations according to

U

(

2πk

N

)

|Q〉 = e2πikQ/N |Q〉 . (2.8)

Each sector is preserved by the gauge–invariant local observables.

Physically, the superselection rule arises because a ZN charge induces quantum–

mechanical effects that can be detected at long range. In particular, this theory

contains a stable cosmic string such that the U(1) gauge field far from the string

satisfies

exp

ie

∮

C

A · dx

= e2πi/N , (2.9)

when integrated over a closed loop C that encloses the string. (The string encloses

6

magnetic flux Φ0/N , where Φ0 = 2π/e is the flux quantum and e is the gauge cou-

pling.) Therefore, the string may be used to measure the ZN charge of a low–energy

projectile in a quantum interference experiment [2,3].

The examples described here illustrate that a superselection rule arises in a gauge

theory whenever certain states are endowed with properties that cannot be destroyed

by any process that is local in spacetime. In this sense, superselection rules provide

a classification of the long–range“hair” that an excitation may carry.

The above discussion can be extended to the case in which the underlying gauge

group G and the discrete subgroup H of G are nonabelian groups. If H is nonabelian,

however, specifying the H charge involves subtleties that we will discuss in sections 7

and 8.

3. The Charge Operator

We will now examine in more detail how the charge of a state may be defined in a

gauge theory with a discrete gauge group. For the sake of concreteness, we continue

to consider the model described in section 2, in which a U(1) gauge symmetry is

broken to ZN . This model is easily generalized.

Defining the ZN charge is delicate, because the η condensate causes the electric

field of a charge to decay as e−mR, where R is the distance from the charge and m is

the photon mass. Thus, the ZN charge cannot be detected classically; it is detectable

at long range only through quantum–mechanical effects that survive in spite of the

screening of the classical electric field.

It was suggested in Ref. [1] that the ZN charge QΣ contained within a closed

surface Σ can be expressed via the Gauss law in terms of a surface integral

exp

(

2πi

NQΣ

)

= exp

2πi

Ne

∫

Σ

E · ds

≡ F (Σ) . (3.1)

Heuristically, one might expect Eq. (3.1) to make sense in spite of the exponential

decay of the classical electric field, since the “flux operator” F (Σ) is unable to see an

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object that carries charge N , and so should be unaffected by the η condensate. The

operator U(2π/N) that represents a ZN gauge transformation could then be defined

as the limit of F (Σ) as the surface Σ tends to spatial infinity. As we will see, this

suggestion is nearly correct, but requires careful interpretation.

Eq. (3.1) is not precisely correct as it stands because quantum mechanical fluc-

tuations cause the expectation value of F (Σ) in a state |ψ〉 to decay rapidly as the

size of Σ increases. We have

limA(Σ)→∞

∣

∣

∣〈ψ|F (Σ)|ψ〉

∣

∣

∣∼ exp[−κA(Σ)] (3.2)

where A(Σ) denotes the area of the surface Σ, and κ is a universal constant, indepen-

dent of the state |ψ〉. This decay of 〈F (Σ)〉 is due to the quantum fluctuations of the

charge–1 field φ. Virtual pairs of charge–1 particles and antiparticles near the surface

Σ cause the charge contained inside Σ to fluctuate. Furthermore, because the photon

is massive, the charge fluctuations near two elements of Σ become very weakly cor-

related when the elements are distantly separated. The finite correlation length thus

gives rise to the characteristic “area–law” decay in Eq. (3.2), and also ensures that

the coefficient κ does not depend on |ψ〉. The area–law behavior of 〈F (Σ)〉 occurs

whether or not the local ZN symmetry is spontaneously broken (by condensation of

φ).

The cases of manifest and broken local ZN symmetry can be distinguished, how-

ever. Although the modulus of 〈F (Σ)〉 exhibits area–law decay, its phase does not get

screened if the ZN symmetry remains manifest. Thus, Eq. (3.1) can be salvaged by

a mere multiplicative rescaling of F (Σ). And since κ is universal, we can isolate the

phase of F (Σ) by dividing by its vacuum expectation value. Thus, we may identify

limA(Σ)→∞

F (Σ)/ 〈F (Σ)〉 = exp

(

2πi

NQ

)

(3.3)

as the operator that represents a global ZN transformation.?

In the remainder of this

? A similar definition of charge was proposed in a related context by Frohlich and Marchetti [9].

8

section and in the following two sections, we will discuss some of the properties of

F (Σ), in order to further elucidate the above arguments.†

We observe first of all that F (Σ) may be regarded as a gauge transformation that

has a discontinuity by the element exp(2πi/N) ∈ ZN on the surface Σ. To see this,

we integrate the Gauss law constraint

∇ · E − eJ0 = 0 (3.4)

against a function ω(x) that vanishes outside Σ and takes the constant value ω =

2π/N inside Σ. An integration by parts then yields

F (Σ) ≡ exp

[

2πi

Ne

∫

Σ

E · ds]

= exp

i

∫

d3x

[

(1

e∇ω

)

·E + ωJ0

]

. (3.5)

Since Ei is the momentum conjugate to Ai, we recognize the right–hand side of

Eq. (3.5) as the operator that represents the gauge transformation with the gauge

function ω(x). Furthermore, since physical states are invariant under arbitrary

smooth local gauge transformations that vanish at spatial infinity, two different gauge

functions ω(x) of compact support with the same discontinuity on Σ define the same

operator F (Σ) acting in the physical Hilbert space. As far as the charge–N η field is

concerned, F (Σ) is a smooth local gauge transformation, and so F (Σ) is incapable of

detecting the fluctuations of η. But the charge–1 φ field can see the discontinuity at

Σ. F (Σ) therefore detects the fluctuations of φ, giving rise to the area–law decay of

〈F (Σ)〉 as described above.

(Now that we recognize F (Σ) as a gauge transformation with a discontinuity on

Σ, it is possible to generalize F (Σ) to the case of an arbitrary underlying (nonabelian)

† In following this discussion, and in interpreting the figures, the reader may find it helpful tothink about the case of three–dimensional Euclidean space; in that case Σ is a one–dimensionalclosed loop.

9

gauge group G. However, F (Σ) is a gauge–invariant operator only if the discontinuity

takes values in the center of the group. The nonabelian case will be further discussed

in section 7.)

It will also be useful to have a Euclidean path integral representation of the

operator F (Σ). In path integral language, a correlation function with an insertion

of F (Σ) is computed by summing over all field configurations such that the gauge

potential Aµ has a “string” singularity on the surface Σ. The string carries “magnetic”

flux 2π/Ne; that is,

exp

ie

∮

C

A · dx

= e2πi/N , (3.6)

where C is an infinitesimal closed loop that encloses the string. (Note that in four–

dimensional Euclidean space, a two–dimensional surface Σ has codimension two, and

so may be enclosed by a one–dimensional loop.)

We may interpret Eq. (3.6) by saying that, when F (Σ) is inserted, the field con-

figurations included in the path integral are restricted to a class that have a ZN dis-

continuity on a (codimension–one) hypersurface Ω that terminates on Σ. A (singular)

gauge transformation can move this hypersurface, and Ω is therefore unphysical. But

gauge transformations cannot move the boundary Σ of Ω, since Σ may be identified

by the gauge–invariant criterion Eq. (3.6)

By this means, we define F (Σ) for an arbitrary closed two–surface Σ. But to

establish the connection with the discontinuous gauge transformation considered pre-

viously, we suppose that Σ lies in the time slice t = 0. Then, by a suitable gauge

choice, we may choose the hypersurface Ω bounded by Σ to lie in the region t ≥ 0, as

indicated in Fig. 1. Because the ZN discontinuity on Ω is absent for t < 0 and present

at t > 0, we can interpret F (Σ) as an operator that creates a ZN discontinuity on Σ.

Thus, for a two–surface Σ lying in a time slice, our path integral formulation of F (Σ)

does indeed coincide with the operator F (Σ) defined by Eq. (3.1).

Of course, whether we use the operator language or the path integral language,

10

our definition of F (Σ) is so far merely formal. To define it rigorously we will need

a short–distance regulator that smooths the discontinuity on Σ. We will describe a

lattice regularization of F (Σ) in section 5.

If the surface Σ does not lie on a time slice, then F (Σ) has a different, and

interesting, interpretation. A generic two–surface Σ intersects a time slice on a one–

dimensional closed loop, and Eq. (3.6) tells us that this loop is the position of a

magnetic flux tube that carries flux Φ = 2π/Ne. An insertion of F (Σ) in the path

integral, then, is equivalent to introducing a cosmic string source that propagates on

the Euclidean worldsheet Σ (Fig. 2).

This observation allows us to connect the operator F (Σ) with the thought ex-

periment described in Ref. [1]. It was emphasized there that the ZN charge of an

object can be measured by scattering the object off a cosmic string that carries mag-

netic flux Φ = 2π/Ne; at low energy, the scattering cross–section is dominated by

the Aharonov–Bohm effect [2, 3]. Equivalently, we can imagine measuring the ZN

charge contained inside a surface Σ by adiabatically winding a cosmic string around

Σ (Fig. 3). The string acquires an Aharonov–Bohm phase exp(2πiQ/N) if Q units

of ZN charge are enclosed. The operator F (Σ) embodies this thought experiment,

since F (Σ) picks up a phase e2πi/N from each ZN charge whose world line crosses the

hypersurface Ω that is bounded by Σ (Fig. 4).

One may gain a greater appreciation of the interpretation of the operator F (Σ)

by contemplating the connection between F (Σ) and the ‘t Hooft loop operator B(C)

defined in Ref. [10]. In the U(1) gauge theory that we have been considering here,

the ‘t Hooft loop operator B(C), acting in a time slice, creates a cosmic string on

the loop C that carries magnetic flux Φ = 2π/Ne. Alternatively, we may think of

this operator as a gauge transformation that has a discontinuity by e2πi/N ∈ ZN on

an open surface Σ such that the boundary of Σ is C . If the charge–N field η were

the only charged field in the theory, then this discontinuity would be a pure gauge

artifact, and the operator B(C) would be independent of the choice of the surface

Σ whose boundary is C . But if there is also a charge–1 field φ, the discontinuity on

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Σ is not merely a gauge artifact; thus B depends on Σ as well as C and should be

denoted B(C,Σ). (In the case studied in Ref. [10], B(C) was dependent on C alone.)

We thus recognize that the flux operator F (Σ) is a degenerate case of the ‘t Hooft

operator B(C,Σ); F (Σ) is obtained by shrinking the loop C to a point, so that the

open surface Σ becomes a closed surface (Fig. 5).

In terms of the ‘t Hooft loop, our formal procedure for measuring the ZN charge

enclosed by the surface Σ may be described in an alternative language. For the sake

of this discussion, let us imagine that all ZN charges are classical sources, so that

there are no quantum fluctuations of the charge. In that event, it is not absolutely

essential to introduce a surface Σ in order to specify the operator B(C,Σ), because

the quantum fields do not see the discontinuity on Σ. However, unless we introduce

such a surface, the ‘t Hooft loop operator B(C) in the presence of classical ZN sources

is typically an N–valued object; because of the Aharonov–Bohm effect, it acquires a

phase e2πi/N upon winding around a unit ZN charge. Since multivalued objects are

awkward to deal with, we may prefer to force B(C) to be single–valued by arbitrarily

restricting it to one of its N branches. The price of single–valuedness is that B(C)

has a cut; we may, for example, select an arbitrary surface Σ bounded by C and

specify that the value of B(C,Σ) jump discontinuously by the multiplicative factor

e2πi/N whenever a unit ZN charge crosses Σ (in the appropriate sense). Although

B(C,Σ) will then be single–valued, its phase will depend on the choice of the surface

Σ.

Now, suppose that we wish to measure the ZN charge inside a closed surface

Σ. If Σ has the topology of a sphere, then we may construct a sequence of loops

Cs, s ∈ [0, 1], each containing a common point P , that sweep through the surface Σ.

The sequence begins and ends with an infinitesimal loop at P (Fig. 6). If B(Cs) is

the multivalued ‘t Hooft loop operator with no cuts, then B(C1) differs from B(C0)

by the phase exp(2πiQΣ/N), where QΣ is the enclosed ZN charge. Alternatively, we

may consider the sequence of single–valued operators B(Cs,∆s) where ∆s is a surface

bounded by Cs that lies on Σ. Since ∆s never crosses any of the ZN charges enclosed

by Σ, B(C1,∆1 = Σ) and B(C0,∆0 = 0) also differ by the phase exp(2πiQΣ/N).

12

But it is evident that

B(C0,Σ) = F (Σ) B(C0, 0) . (3.7)

As the loop C sweeps out the surface Σ, it drags behind it the surface on which the

‘t Hooft loop has its cut. When C shrinks back to a point, only the cut on Σ is left

behind. So we may think of F (Σ) as a measure of the multivaluedness of the ‘t Hooft

loop (at least when there are no quantum–mechanical charge fluctuations).

Expressed in this language, our procedure for measuring the ZN charge is strongly

reminiscent of a closely analogous procedure that has been discussed previously. Cole-

man [11] and Srednicki and Susskind [12] considered the problem of measuring a ZN

magnetic charge in the confining phase of an SU(N) Yang–Mills theory. They noted

that the confining phase has a mass gap, so that magnetic fields are screened and

magnetic charge cannot be detected classically. But they claimed that the ZN mag-

netic charge induces quantum–mechanical effects that can be detected at long range.

Specifically, they noted that the Wilson loop operator is multivalued in the presence

of ZN magnetic charges, and that the ZN magnetic charge enclosed by a surface Σ can

be detected as the phase acquired by a Wilson loop that winds around Σ. (Monopoles

were treated as classical sources in Refs. [11] and [12], and no attempt was made to

take into account quantum–mechanical magnetic charge fluctuations.)

This analogy between the detection of ZN electric charge and the detection of ZN

magnetic charge can be made precise. In 3 + 1 dimensions, a duality transformation

can be formulated [10] that interchanges electric and magnetic charge, and hence

also interchanges the Wilson loop operator and the ‘t Hooft loop operator. Duality

thus relates confining behavior, in which the expectation value of the Wilson loop

decays according to an area law and magnetic fields are screened, to Higgs behavior,

in which the expectation value of the ‘t Hooft loop decays according to an area law

and electric fields are screened. In a confining theory, the phase of a large Wilson

loop is able to respond to the long range field of a magnetic monopole, even though

magnetic screening causes the field to decay exponentially, just because confinement

also causes the modulus of the Wilson loop to decay rapidly [11]. Likewise, the rapid

13

decay of the modulus of the ‘t Hooft loop in a Higgs theory enables the phase of the

‘t Hooft loop to respond to the weak long range field of an electric charge.

We see, then, that the detection of a ZN electric charge by an ‘t Hooft loop and

of a ZN magnetic charge by a Wilson loop involve essentially the same mathematics.

In both cases, the crucial feature is that a unit ZN electric charge can see the Dirac

string of a ZN magnetic charge through a nontrivial Aharonov–Bohm effect.

In fact, the argument of Ref. [11] and [12] shows that black holes can in principle

carry magnetic quantum–mechanical hair, as well as the electric quantum–mechanical

hair proposed in Ref. [1]. In an SU(N) gauge theory, ZN magnetic monopoles are

consistent with the Dirac quantization condition only if all matter fields are invari-

ant under ZN , the center of the group. These monopoles are confined by magnetic

flux tubes in the Higgs phase, but in the confinement phase there is a nontrivial ZN

magnetic charge superselection rule. The ZN magnetic charge of an object can be de-

tected at long range, because a monopole has an Aharonov–Bohm interaction with an

electric flux tube. (Were we to push the analogy with the electric superselection rule

even further, we would distinguish two types of confinement phase. If ZN monopoles

condense, then ZN magnetic charge is screened even quantum mechanically, and the

electric flux tube becomes the boundary of a ZN domain wall [13].)

However, in the realistic case of an [SU(3)color× U(1)em]/Z3 gauge theory, we may

not legitimately speak of quantum–mechanical hair. Though magnetic monopoles

carry a Z3 color magnetic flux, there are no stable electric flux tubes, and hence

no means of detecting quantum–mechanical hair. The U(1)em magnetic charge is the

only magnetic quantum number that can be detected at long range, and the magnetic

hair is entirely classical. If black holes can carry magnetic quantum–mechanical hair

in Nature, then, this hair is not associated with the known strong interaction; rather,

it must be associated with another, as yet unknown, confining gauge interaction that

admits genuine ZN monopoles.

Incidentally, much as the ZN magnetic monopole number respects a superselec-

tion rule in a confining 3 + 1–dimensional gauge theory, so the ZN vortex number

14

respects a superselection rule in a 2+1–dimensional theory with manifest local ZN

symmetry. The ZN vortex number can be detected by Aharonov–Bohm scattering

off a ZN electric charge. Although a gauge–invariant operator can be constructed

that annihilates a vortex, this operator is not local; it has a semi–infinite string

that can be seen by matter fields with nonvanishing ZN charge. This string is the

(2+1)–dimensional analog of the surface that, in 3+1 dimensions, stretches across the

‘t Hooft loop. If the local ZN symmetry is spontaneously broken, then the vortices are

confined. Hence, neither the vortex superselection rule nor the charge superselection

rule survives.

We have now formulated the notion of quantum–mechanical hair in a reasonably

precise language. To conclude this discussion, we wish to compare the quantum–

mechanical hair associated with a local discrete symmetry to another exotic type of

hair that was proposed recently. Bowick et al. [14] considered a theory in which a

global U(1) symmetry is spontaneously broken. Such a theory contains an exactly

massless Goldstone boson, the axion, and also a topological defect, the axion string.

An axionic charge operator can be defined, and an object that carries this charge

exhibits a nontrivial Aharonov–Bohm effect with an axion string. By means of this

Aharonov–Bohm effect, axionic charge can in principle be detected at long range;

thus, axionic charge is a type of hair.

Since axionic charge is detected via an Aharonov–Bohm interaction with a global

string, much as ZN electric charge is detected via an Aharonov–Bohm interaction

with a gauge string, these two types of hair appear to be related. Actually, axionic

charge is more akin to the ZN vortex number that arises in a discrete gauge theory

in 2+1 dimensions than to ZN electric charge. To see this connection more clearly,

it is helpful to recognize that the massless axion field Θ is dual to a three–form field

strength H defined by

H =

(

f2

2π

)

∗ dΘ , (3.8)

where ∗ denotes the Hodge dual. (We have normalized Θ so that it is a dimen-

sionless periodic variable with period 2π; f is the mass scale that characterizes the

15

spontaneous breakdown of the global U(1) symmetry.) This field strength H can be

expressed as the curl of a two–form potential B,

H = dB , (3.9)

and the axionic charge q in a volume Ω that is enclosed by the surface Σ can be

expressed as a surface integral

q =1

2π

∫

Ω

H =1

2π

∫

Σ

B . (3.10)

The quantity q becomes more recognizable when we re–enact this duality trans-

formation in 2+1 dimensions. Then the axion field is dual to an electromagnetic

two–form F , and the axionic charge in a region Σ that is enclosed by the loop C is

q =e

2π

∫

Σ

F =e

2π

∫

C

A ; (3.11)

it is just the magnetic flux in Σ, in units of the flux quantum. Furthermore, axionic

charge is detected in 2+1 dimensions via an Aharonov–Bohm interaction with an

axion vortex, and the vortex is transformed under duality into an electrically charged

particle. Duality, then, maps the detection of axionic charge by an axion vortex, in

2+1 dimensions, to the detection, in (2+1)–dimensional electrodynamics, of magnetic

flux with an electrically charged particle. (As in electrodynamics, the Aharonov–

Bohm interaction is only sensitive to the axionic charge modulo an integer.)

It is also enlightening to consider the fate of the axionic hair when the dual

electrodynamics is in a phase other than the Coulomb (massless) phase. For example,

we may include finite–action magnetic monopole configurations in the Euclidean path

integral of (2+1)–dimensional electrodynamics [15]. Then the theory acquires a mass

gap, and electric charges become confined by stable electric flux tubes. In the dual

description in terms of the axion field, the magnetic monopoles break the global U(1)

16

symmetry intrinsically [13]. Hence the axion acquires a nonzero mass and the axion

vortex becomes the boundary of an axion domain wall; the domain wall is dual to

the electric flux tube. In this confining phase, it is inappropriate to regard axionic

charge as a type of hair that can be detected at long range. Instead, the axion vortex

detects the axionic charge by dragging a domain wall across the charge.

If (2+1)–dimensional electrodynamics is in the Higgs phase, then there is a Meiss-

ner effect, and, consequently, magnetic flux is quantized. The axionic charge, or mag-

netic flux, contained in an isolated region cannot assume an arbitrary value, but must

be an integer multiple of the flux quantum. If the electromagnetic gauge invariance

is completely broken, then a quantum of magnetic flux cannot be detected via the

Aharonov–Bohm effect. But if there is a surviving ZN local symmetry, then there is

a ZN vortex number that can be detected at long range. In this sense, ZN vortex

hair is the remnant of axionic hair that may survive the Higgs mechanism, in 2+1

dimensions.

In the dual description in terms of the axion field, the Higgs mechanism corre-

sponds to the restoration of the global U(1) symmetry due to the condensation of

axion vortices [13]. ZN–valued axionic hair survives the Higgs mechanism if the vor-

tex that condenses is not the minimal axion vortex with unit winding number, but

rather a nonminimal vortex with winding number N .

This discussion of the (2+1)–dimensional cases is readily generalized to the de-

tection of axionic charge by axion strings, in 3+1 dimensions. If the axion is exactly

massless, then the axionic charge q may assume any value, and q modulo an inte-

ger can be detected via the Aharonov–Bohm interaction of the charge with an axion

string. If the U(1) global symmetry is intrinsically broken, then the axion has a

nonzero mass and the axion string is the boundary of an axion domain wall. An

axion string detects charge by dragging a domain wall across the region that contains

the charge. If the U(1) global symmetry is manifest, then axion strings condense

[16] and the axionic charge is quantized. If the strings that condense are nonminimal

strings with winding number N , then axionic hair takes values in ZN . The ZN–valued

17

axionic charge can be detected at long range by an axion string with winding number

one.

It is convenient to describe axion physics in terms of the two–form potential B,

because axionic hair can then be discussed in the language of classical field theory,

whether or not the global U(1) symmetry is spontaneously broken. An object that

carries axionic charge has a long–range B field. Now, B is not itself a gauge–invariant

quantity, and the long–range B field is actually a pure gauge locally, but the gauge–

invariant axionic charge q can be expressed as a surface integral of B, as in eq. (3.10).

The statement that B may be regarded as a “classical” field should not be misinter-

preted. This statement means that quantum fluctuations of B can be neglected. But

B is not a classical local observable, and the “topological” charge q can be detected

at long–range only via quantum–mechanical interference effects.?

This “classical” description of axionic hair is exploited by Bowick et al. [14] in

their analysis of axionic black holes. They note that the classical field equations for

B coupled to Einstein gravity admit black hole solutions with nonzero axionic charge

q. Furthermore, these axionic black holes obey a generalized uniqueness theorem. On

a stationary black hole with a nonsingular event horizon, the potential B is required

to be a pure gauge locally, except at the singularity. Thus, the axionic charge q is

the sole physical attribute that characterizes the axion field of a stationary black hole

[14].

A black hole that carries axionic charge contrasts sharply with a black hole that

carries ZN electric charge. In classical field theory, black hole uniqueness theorems

[17] require massive scalar and vector fields to vanish exactly on a stationary black

hole with a nonsingular event horizon. In a Higgs phase, then, a stationary black

hole can carry no classical electric or scalar charge. Hence, the quantum–mechanical

correlations that can be detected far from the black hole resist being encoded in any

classical field–theoretic description. In this respect, ZN electric hair is a more subtle

? In fact, a nonsingular gauge transformation on the surface Σ can change q by an integer; thatis why only q modulo an integer can be detected at long range.

18

and elusive notion than axionic hair. The manifestation of these quantum–mechanical

correlations during gravitational collapse, and their effect on the subsequent evapo-

ration of a charged black hole, may be worthy of further study.

4. The Order Parameter

Our discussion in section 2 indicated that, in the U(1) gauge theory with charge–

N field η and charge–1 field φ, there are two distinct Higgs realizations of the gauge

symmetry. These two realizations may be distinguished according to whether a ZN

subgroup of U(1) remains manifest, or in other words, whether the theory respects a

ZN superselection rule. If both realizations can be achieved for a suitable choice of

the parameters in the model, then we expect that there are two distinct Higgs phases,

separated by a phase boundary. In this section, we propose an order parameter that

is sensitive to such a phase transition. Our order parameter is readily generalized to

one that probes the realization of an arbitrary abelian discrete gauge symmetry.

Heuristically, whether the charge–1 field φ “condenses” determines whether the

ZN symmetry is manifest or spontaneously broken. But Elitzur’s theorem [18] cau-

tions us that a gauge non–invariant local order parameter is unable to reveal the

nontrivial phase structure. Our order parameter must instead be a gauge–invariant

and nonlocal object.

It is familiar that such nonlocal order parameters can distinguish the Higgs real-

ization from the confining realization of a gauge symmetry. Physically, the confining

phase supports stable electric flux tubes and the Higgs phase supports stable magnetic

flux tubes. (Both phases are distinguished from the Coulomb phase in that there is a

mass gap, and, in the case of a Higgs phase in which the gauge symmetry is completely

broken, no nontrivial charge superselection rule.) Mathematically, appropriate order

parameters are the Wilson loop [19] and ‘t Hooft loop [10] operators.

Suppose, for example, that the gauge group is G = SU(N) and that all of the

fields in the theory transform trivially under the center ZN of SU(N). The Wilson

19

loop

W (C) = tr

[

P exp

(

ig

∮

C

A · dx)]

(4.1)

may be regarded as an insertion of a classical source, transforming as the defining

representation of SU(N), that propagates along the worldline C . If electric flux is

confined, then C becomes the boundary of the worldsheet of an electric flux tube; for

sufficiently large loops, W (C) therefore exhibits the area–law behavior

〈W (C)〉 ∼ exp[−κA(C)] , (4.2)

where A(C) is the minimal area of a surface bounded by C , and κ is the string

tension. In the Higgs phase, electric flux is screened, and the Wilson loop has the

perimeter–law behavior

〈W (C)〉 ∼ exp[−µ P (C)] , (4.3)

where P (C) is the length of C . Conversely, the ‘t Hooft loop B(C) may be regarded

as an insertion of a classical ZN magnetic monopole source that propagates along

the world line C . (That is, C is the boundary of a ZN Dirac string, as described

in section 3.) In the Higgs phase, magnetic flux is confined and B(C) has area–law

behavior, while in the confinement phase, magnetic flux is screened and B(C) has

perimeter–law behavior.

But if an SU(N) gauge theory is coupled to “quark” matter in the defining rep-

resentation of SU(N) (or in any representation that transforms faithfully under the

center ZN ), then the confining and Higgs realizations can no longer be distinguished

by the above criteria. Quark–antiquark pairs appear as quantum fluctuations, al-

lowing the electric flux tube to break. The Wilson loop therefore always obeys the

perimeter law. Because quarks transform nontrivially under ZN , the ‘t Hooft loop

becomes B(C,Σ); it depends on the choice of the surface Σ bounded by C as discussed

in section 3, and always obeys the area law.

20

With W (C) and B(C) failing to distinguish the Higgs and confining phases, one

recognizes that no such distinction may be possible when “quarks” are present; there

need be no sharp phase boundary that separates the Higgs and confining regions of

the phase diagram [10,20,21]. However, we have also argued that two types of Higgs

phases are possible, depending on the realization of the local ZN symmetry. It is

presumably the Higgs phase with spontaneously broken ZN symmetry, and hence no

nontrivial superselection rule, that is indistinguishable from the confining phase.

How, though, do we distinguish the two types of Higgs phases? Topological

defects provide a potentially useful criterion. The Higgs phase with manifest ZN

gauge symmetry supports cosmic strings that carry one unit of ZN magnetic flux.

But when the local ZN symmetry is spontaneously broken, such a string becomes

the boundary of a domain wall [22-24]. This observation might tempt one to propose

that the behavior of 〈F (Σ)〉 distinguishes the two phases. We have seen that F (Σ)

may be regarded as an insertion of a classical ZN cosmic string source propagating on

a worldsheet Σ; if Σ becomes the boundary of a domain wall, then we might expect

〈F (Σ)〉 to decay like exp[− Volume]. Unfortunately, this proposal does not quite

work, because the domain wall bounded by Σ is not absolutely stable. A hole in

the wall, bounded by a ZN string, can arise as a quantum fluctuation. A sufficiently

large hole will grow catastrophically, devouring the wall (Fig. 7). Thus, even if the

wall is very long–lived, 〈F (Σ)〉 will always decay like exp[− Area] for sufficiently large

surfaces and there is no need for 〈F (Σ)〉 to behave non–analytically at the boundary

between phases with manifest and broken local ZN symmetry.

The preferred way to distinguish the two phases is by means of the ZN charge

superselection rule, as described in section 2. If the local ZN symmetry is manifest,

then ZN electric charge is screened classically but not quantum mechanically; ZN

charge can be detected at long range via the Aharonov–Bohm effect. If the local ZN

symmetry is spontaneously broken, then ZN electric charge is completely screened.

We may imagine introducing a classical source of ZN charge at the origin and then at-

tempting to detect the charge at spatial infinity. The charge is detectable in principle

if and only if the local ZN symmetry is manifest.

21

To restate this criterion mathematically, we recall that a Wilson loop operator

W (C) acts as a source of ZN charge, and that the flux operator F (Σ) can detect ZN

charge, as described in section 3. If we define

A(Σ, C) =〈F (Σ) W (C)〉〈F (Σ)〉 〈W (C)〉 , (4.4)

then, if ZN electric charge is unscreened, we have

lim A(Σ, C) = exp

[

2πi

Nk(Σ, C)

]

. (4.5)

Here the limit is taken with Σ and C increasing to infinite size, and with the closest

approach of Σ to C also approaching infinity; k(Σ, C) denotes the (integer–valued)

linking number of the surface Σ and loop C . That is, k(Σ, C) is the (signed) number

of times the world line C crosses a volume Ω that is bounded by Σ (Figure 8). If,

however, ZN electric charge is screened, then

lim A(Σ, C) = 1 . (4.6)

The nonanalytic behavior of A(Σ, C) guarantees that the two Higgs phases are sepa-

rated by a well–defined phase boundary.

Our order parameterA(Σ, C) can obviously be generalized to probe the realization

of any abelian local discrete symmetry. In fact, we will see in section 6 that A(Σ, C)

can be constructed without any explicit reference to a continuous gauge group in

which the discrete gauge group is embedded.

5. Example: Lattice Theory with Coupled Z2 Gauge and Z2 Spin Variables

Since the above discussion is rather abstract, one desires an explicit model in

which the behavior of the order parameter A(Σ, C) can be studied analytically. We

now present such an example. Our model is a Z2 lattice gauge theory coupled to a Z2

spin system; the spin systems plays the role of “matter” that transforms nontrivially

22

under the local discrete Z2 symmetry [20,25,26]. The virtue of this model is that both

its weak and strong coupling behavior can be analyzed using convergent expansions.

Thus, the phase diagram of the model can be mapped out using perturbation theory.

We will indeed find a phase boundary that separates a phase with screened Z2 charge

from a phase with unscreened Z2 charge. In spite of the simplicity of the example,

we believe that it captures all of the essential features of the general case.

The degrees of freedom of the model are gauge variables

U` ∈ Z2 ≡ 1,−1 (5.1)

residing on links (labeled by `) of a cubic four–dimensional spacetime lattice, and

spin variables

φi ∈ Z2 ≡ 1,−1 , (5.2)

residing on sites (labeled by i). The Euclidean action is

S = Sgauge + Sspin ,

where

Sgauge = −β∑

P

UP , (5.4)

and

Sspin = −γ∑

`

(φUφ)` . (5.5)

Here UP =∏

`∈P

U` associates with each elementary plaquette (labeled by P ), the

product of the four U`’s associated with the links of the plaquette, and (φUφ)ij =

φiUijφj , for each pair ij of nearest neighbor sites. The action is invariant under the

23

Z2 gauge transformation defined by

ηi ∈ Z2 ≡ 1,−1 , (5.6)

where the variables transform as

φi → ηiφi , Uij → ηiUijηj . (5.7)

Expectation values of gauge–invariant quantities are computed using the normalized

probability measure Z−1 e−S, where

Z =∑

Uφ

e−S . (5.8)

Notice that, under a nontrivial global gauge transformation ηi = −1, the gauge

variable U` is invariant but the spin variable φi is not. This model is therefore a

prototype of the phenomenon that we are investigating; the “matter” field φ carries

the local discrete Z2 charge, and we wish to determine whether there is a nontrivial

Z2 superselection rule.

Our model is tractable because it can be analyzed by means of convergent per-

turbation expansions if both coupling parameters β and γ are either large or small.

We will show by applying the order parameter A(Σ, C) that Z2 charge is unscreened

if β is large and γ is small, and that Z2 charge is screened if either β is small or γ

is large. This nonanalytic behavior of A(Σ, C) establishes the existence of a phase

boundary.

It is easy to anticipate this conclusion if one contemplates the limiting behavior of

the model on the boundaries of the phase diagram (Fig. 9). The model becomes trivial

in the limits β = 0 and γ = ∞. For β = 0, the UP ’s are completely unconstrained,

and the U`’s are therefore free to adjust so that (φUφ)` = 1 on every link, whatever

the configuration of spins. For γ = ∞, (φUφ)` = 1 must be satisfied on every link,

and hence UP =∏

`∈PU` =

∏

`∈P(φUφ)` = 1 on every plaquette; there is no dependence

on β.

24

More interesting are the limits β = ∞ and γ = 0. For β = ∞, the gauge variables

are frozen at U` = 1 (up to a gauge transformation) and the model reduces to a Z2

Ising spin system. There is thus a second–order phase transition at a critical coupling

γ = γc on the β = ∞ axis; the spins are disordered for γ < γc and ordered for γ > γc.

For γ = 0, the spins decouple, and the model reduces to a Z2 gauge system. There is

thus a first–order phase transition (in four or more Euclidean dimensions) at a critical

coupling β = βc on the γ = 0 axis; the gauge system is in the confining phase for

β < βc and in the Higgs phase for β > βc.

In the region of the phase diagram with β large and γ small, then, we expect a

Higgs phase with disordered (uncondensed) spins. This region, in which Z2 charge in

unscreened, should be separated by a phase boundary from the large γ and small β

regions. We therefore expect the phase diagram of the model to have the schematic

form suggested in Fig. 9.

This phase structure has been conjectured previously [20,25], and has been con-

firmed by Monte Carlo simulations [27]. Various attempts have been made to identify

an order parameter that distinguishes the two phases. (These attempts are reviewed

in Ref. [7].) To our knowledge, though, no order parameter has been suggested before

that probes directly whether Z2 charge is screened, as A(Σ, C) does. We will confirm

the phase structure indicated in Fig. 9 by computing A(Σ, C) explicitly.

To begin, we must construct a lattice version of the flux operator F (Σ). To do

this, we consider Σ to be a closed surface made up of plaquettes of the dual lattice;

then each plaquette of Σ is dual to a plaquette of the original lattice (Fig. 10).?

To

evaluate the path integral with an insertion of F (Σ), we perform the transformation

UP → −UP , P ∈ Σ∗ (5.9)

on these plaquettes that are dual to the plaquettes of Σ. That is, we flip the sign of

UP , or, equivalently, flip the sign of β on these plaquettes, frustrating them. In effect,

? This construction is easier to visualize in three–dimensional Euclidean space. Then Σ is aclosed loop made up of links of the dual lattice, and each link of Σ is dual to a plaquette ofthe original lattice.

25

we place a unit of Z2 “magnetic flux” on the surface Σ, so that Σ can be regarded as

the worldsheet of a Z2 Dirac string.†

Before we proceed to the case of a Z2 gauge system coupled to matter, consider

first the pure gauge system. We noted that F (Σ) is merely a gauge transformation in

a pure gauge theory, and the surface Σ is an artifact that can be moved by means of a

singular gauge transformation. The lattice analog of a singular gauge transformation

is the change of variable [28]

U` → −U` . (5.10)

By performing this change of variable on a link contained in one of the plaquettes

that is dual to Σ, we can move the surface Σ. In fact, Σ is the boundary of a set of

cubes of the dual lattice, and by performing U` → −U` on each of the links that are

dual to the cubes of this set, we can shrink Σ to a point.‡

So it is evident that F (Σ)

is a mere change of variable, and that

〈F (Σ)〉 = 1 . (5.10)

It is not the case, however, that F (Σ) can be replaced by 1 when it is inserted in

an arbitrary Green function. The Wilson loop on the lattice is defined as

W (C) =∏

`∈C

U` (5.12)

where C is a closed loop of links. Now we easily see that

〈F (Σ)W (C)〉 = −〈W (C)〉 , (5.13)

if the loop C and surface Σ have linking number 1 (or any odd integer). This is

because F (Σ) is a change of variable that flips the sign of U` on an odd number of

† This method of introducing a Dirac string on the lattice was used in Ref. [28].‡ In three–dimensional Euclidean space, Σ is the boundary of a set of plaquettes of the dual

lattice, and we shrink Σ to a point by performing U` → −U` on each of the links dual to theseplaquettes.

26

the links of C . Eq. (5.13) is the statement that Z2 charge is not screened in the pure

gauge system, or, equivalently, that a Z2 cosmic string can be detected at long range

by a Z2 charge. In the pure gauge system, this statement is purely kinematic; it

has nontrivial dynamical content only if fluctuating matter fields are introduced that

carry the Z2 charge.

It is somewhat enlightening nonetheless to consider how Eq. (5.13) is fulfilled in

the confining and Higgs phases of the pure gauge model. In the confining phase, we

formulate a strong coupling expansion by reexpressing [26]

e−Sgauge = N(β)∏

P

(1 + UP tanhβ) , (5.14)

and expanding in powers of tanhβ. Nonvanishing contributions to 〈W (C)〉 are associ-

ated with surfaces bounded by C that are “tiled” by extracting a factor of UP tanhβ

from Eq. (5.14) for each plaquette of the surface. Loosely speaking (because there

are contributions from disconnected surfaces as well) a surface containing A plaque-

ttes makes a contribution of order (tanhβ)A. The number of such surfaces grows

sufficiently slowly with A that the strong–coupling expansion has a finite radius of

convergence [29]. Thus, for a planar loop, and sufficiently small tanhβ, we have

〈W (C)〉 = (tanhβ)A + . . . (5.15)

where A is the area of the minimal surface bounded by C , and the remainder is

negligibly small. Eq. (5.15) shows that the Z2 gauge theory exhibits confinement at

strong coupling.

If Σ and C have an odd linking number, then

〈F (Σ)W (C)〉 = − (tanhβ)A + . . . , (5.16)

because the minimal surface bounded by C contains an odd number of plaquettes on

which tanhβ has flipped sign. Heuristically, a source of Z2 charge can “see” the Dirac

string because it drags along an electric flux tube (represented by the minimal–area

surface) that crosses the string at some point [12].

27

In the Higgs phase, a weak–coupling expansion in e−2β can be carried out, where

the order in the expansion is determined by the number of “frustrated” plaquettes

with UP = −1. Since there is a duality transformation that interchanges strong

and weak coupling in this model [26,28], the weak–coupling expansion also has a

finite radius of convergence. At weak coupling, the leading nontrivial contribution

to 〈W (C)〉 arises when one of the links ` on C assumes the value U` = −1; this

configuration has W (C) = −1 and six frustrated plaquettes. Thus,

〈W (C)〉 ∼ exp(

− L(e−2β)6 + . . .)

exp(

L(e−2β)6 + . . .) = exp

(

− 2L(e−2β)6 + . . .)

(5.17)

where L is the number of links on C . The exponential in the numerator of Eq. (5.17)

results from summing over the LN/N ! ways of flipping the sign on N links contained

in C ; the denominator is the contribution from these configurations to the partition

function Z = Σe−S. Eq. (5.17) shows that the Z2 gauge theory is not confining at

weak coupling.

The leading contribution to 〈F (Σ)W (C)〉 at weak coupling arises from a configu-

ration of the link variables such that none of the plaquettes dual to Σ are frustrated.?

In order to avoid frustrating any plaquettes, we must choose U` = −1 on all of the

links that are dual to the cubes enclosed by Σ.†

This is the sense in which, in the

Higgs phase, the cosmic string represented by Σ has “hair” that can be detected at

long range; F (Σ) evidently flips the sign of W (C) if Σ and C have an odd linking

number.

Let us now turn to the less trivial case of coupled Z2 gauge and spin systems.

As for the gauge coupling β, the dependence on the spin coupling γ can be studied

by means of strong–coupling and weak–coupling expansions. At strong coupling, we

? We say that a plaquette is frustrated if the plaquette action is not at its minimum. Thus, whenF (Σ) is inserted in a Green function, a plaquette P dual to Σ is frustrated if UP = +1, whilea plaquette P not dual to Σ is frustrated if UP = −1.

† In three–dimensional Euclidean space, we choose U` = −1 on all of the links that are dual tothe plaquettes enclosed by Σ.

28

write

e−Sspin = N(γ)∏

`

[

1 + (φUφ)` tanhγ]

(5.18)

and expand in powers of tanhγ. At weak coupling, we expand in powers of e−2γ, with

a factor of e−2γ arising from each frustrated link with (φUφ)` = −1. The strong–

coupling expansion has a finite radius of convergence because the number of closed

loops of length L grows sufficiently slowly with L; the weak–coupling expansion can

be seen to have a finite radius of convergence by a duality argument.

We now distinguish four cases:

(i) β, γ << 1

In this region, the Wilson loop exhibits perimeter law behavior. For a sufficiently

large loop C , the leading contribution to 〈W (C)〉 arises when a factor of (φUφ)` tanhγ

is extracted from Eq. (5.18) for each link of C . Thus

〈W (C)〉 = (tanhγ)L + . . . , (5.19)

where L is the length of C . The interpretation is clear: the theory confines Z2 charge,

but the electric flux tube can break. W (C) creates a Z2–invariant “hadron,” rather

than an isolated source of Z2 charge.

To determine the leading nontrivial behavior of 〈F (Σ)〉, consider the effect on the

partition function Z of changing the sign of β on the plaquettes dual to Σ. Since a

contribution to Z of order (tanhγ)4 tanhβ arises from tiling a single plaquette with

UP tanhβ and covering each link of the plaquette with (φUφ)`, we find that

〈F (Σ)〉 =exp[−A(tanhγ)4tanhβ + . . .]

exp[A(tanhγ)4tanhβ + . . .]

= exp[−2A(tanhγ)4tanhβ + . . .] , (5.20)

where A is the number of plaquettes of Σ. (The exponentiation results from summing

over the AN/N ! ways of tiling N of the plaquettes dual to Σ.) The interpretation is

29

again clear: 〈F (Σ)〉 has area–law behavior because virtual pairs of Z2 charges cause

uncorrelated fluctuations in the total Z2 charge enclosed by Σ, as we discussed in

section 3.

It is quite evident that

lim A(Σ, C) = lim〈F (Σ)W (C)〉〈F (Σ)〉 〈W (C)〉 = 1 . (5.21)

Since the leading behavior of W (C) is zeroth–order in tanhβ, it is completely un-

affected by changing the sign of β on the plaquettes dual to Σ. Only nonleading

contributions to 〈W (C)〉 that decay like exp[−Area] are affected by F (Σ) if Σ and

C are far apart. Because of confinement, there are no free Z2 charges, and it is not

possible to detect Z2 charge at long range.

(ii) β, γ >> 1

Much as in the pure gauge theory, the leading nontrivial contribution to 〈W (C)〉at weak coupling arises when one of the links on C has U` = −1, except that now

flipping U` frustrates the spins on the link as well as the six plaquettes that contain

the link. Thus, we find

〈W (C)〉 = exp[−2L(e−2β)6 e−2γ + . . .] , (5.22)

where L is the length of the loop C .

The flux operator F (Σ) frustrates the plaquettes dual to Σ, and so its leading

behavior is

〈F (Σ)〉 = (e−2β)A + . . . , (5.23)

where A is the area of Σ. Eq. (5.23) is actually the correct leading weak–coupling

behavior only for sufficiently large surfaces, where sufficiently large means, roughly

speaking,

(e−2γ)A12 <∼ e−2β . (5.24)

If Eq. (5.24) is not satisfied, then we can do better than Eq. (5.23) by flipping a set

of links so as to shrink the surface Σ dual to the frustrated plaquettes to a surface

30

Σ of smaller area A < A. Shrinking the surface saves some factors of e−2β, but at

the price of new factors of e−2γ that result from frustrating the spins on the links

that are dual to the volume V enclosed between Σ and Σ. Since shrinking the surface

by δA requires that the spins be frustrated in a volume δV ∼ A1

2 δA, it becomes

disadvantageous to shrink Σ when Eq. (5.24) is satisfied.

We can see again that

lim A(Σ, C) = 1 , (5.25)

because, for Σ sufficiently large, the leading contribution to 〈F (Σ)〉 does not require

U` to flip sign on links that are deep inside the surface bounded by Σ. A cosmic

string has no hair because hair is too costly; the action due to the hair scales like the

volume enclosed by the worldsheet of the string. This is just the phenomenon noted

in section 3 — condensation of the matter field causes the cosmic string to become

the boundary of a domain wall, but the wall is unstable and decays by nucleation of

a loop of string.

(iii) β << 1, γ >> 1

The leading nontrivial contribution to 〈W (C)〉 is zeroth order in tanhβ and arises,

again, when one link on C is flipped in sign. This frustrates the spins on that link,

so that we find

〈W (C)〉 = exp(−2L e−2γ + . . .) . (5.26)

where L is the length of C .

The leading contribution to 〈F (Σ)〉 is zeroth–order in e−2γ. In the γ → ∞ limit

all plaquette variables are frozen at UP = 1. By considering the effect on the

partition function of changing the sign of β on the plaquettes dual to Σ, we therefore

find

〈F (Σ)〉 = exp(−2A tanhβ + . . .) , (5.27)

where A is the area of Σ.

31

It is obvious again that

lim A(Σ, C) = 1 . (5.28)

Z2 charge is both confined and screened by the spin condensate.

(iv) β >> 1, γ << 1

The leading nontrivial contribution to 〈W (C)〉 arises in zeroth order in tanhγ,

and we therefore have

〈W (C)〉 = exp[

− 2L(e−2β)6 + . . .]

, (5.29)

just as in the weakly–coupled pure gauge theory.

There is a contribution to 〈F (Σ)〉 of the form (e−2β)A that arises when all of the

plaquettes dual to Σ are frustrated, just as in case (ii) above. But now a much larger

contribution is obtained by flipping all of the links dual to the volume enclosed by

Σ. Then UP = −1 on the plaquettes dual to Σ and UP = 1 elsewhere, so that no

plaquette variables are frustrated. By expanding the spin partition function with the

plaquette variables frozen at these values, we find

〈F (Σ)〉 = exp[

− 2A(tanhγ)4 + . . .]

. (5.30)

The crucial feature is that the configurations that dominate 〈F (Σ)〉 have the gauge

variables U` flipped in a volume enclosed by Σ. Thus, a cosmic string has hair, and

we have shown that

limA(Σ, C) = −1 (5.31)

if Σ and C have an odd linking number.

We have therefore established that the model is in a phase with unscreened Z2

charge for β >> 1 and γ << 1, and that A(Σ, C) serves as an order parameter for

the phase transition.

32

It is obvious that this order parameter A(Σ, C) can be generalized to a lattice

gauge theory with arbitrary gauge group G, when the discrete symmetry whose real-

ization is to be probed is in the center of G.

6. Local Discrete Symmetry without Gauge Fields

We have seen how a field theory that respects a local discrete symmetry can arise

by means of the Higgs mechanism as the low–energy limit of an underlying theory

with a continuous gauge group. We wish to explain in this section how a theory with

a discrete gauge symmetry can be formulated directly, without ever introducing any

gauge fields. We will see that the order parameter defined in section 4 can still be

constructed in order to probe whether a nontrivial superselection rule is respected by

the theory.

The discussion in this section will be somewhat formal, however, in that, although

we will use notation appropriate for a continuum field theory, a finite ultraviolet cutoff

will be implicit, and a nontrivial continuum limit of the discrete gauge theory need

not necessarily exist. Actually, it is obvious that a discrete gauge theory without

gauge fields can be constructed, for we can integrate out any gauge fields that acquire

mass by the Higgs mechanism, thus obtaining an effective field theory with a cutoff

[30]. But it is convenient and rather enlightening to describe this effective field theory

directly, without any reference to the physics at mass scales above the cutoff.

To keep the discussion concrete, we consider again the U(1) gauge theory de-

scribed previously, with a charge–N scalar field η, and a charge–1 scalar field φ. (As

usual, the generalization is straightforward.) We imagine that η condenses at the

mass scale v, so that the photon acquires mass µ = Nev. We may integrate out η

and the photon to obtain an effective field theory for the surviving field φ, with cutoff

Λ ∼ µ.

For the purpose of writing the Lagrangian of this theory, it is convenient to retain

the pure gauge Goldstone degree of freedom, the phase of η, that is eaten by the

33

photon. We denote this phase by Θ, where

η = ρ e−iΘ ; (6.1)

thus Θ is a periodic variable defined modulo 2π. Under a U(1) gauge transformation

parametrized by ω, the fields φ and Θ transform according to

φ→ eiω φ , Θ → Θ −Nω . (6.2)

The effective Lagrangian with (nonlinearly realized) local U(1) invariance is [30]

L(φ,Θ) = L(φ eiΘ/N )

= ∂µ(φ eiΘ/N)† ∂µ(φ eiΘ/N )

− V (φ eiΘ/N ) + . . . , (6.3)

where the ellipsis indicates terms higher–order in derivatives. That is, the action is a

local functional of the U(1) invariant field

Φ = φ eiΘ/N . (6.4)

Furthermore, since the ZN transformation

Φ → e2πi/N Φ (6.5)

is merely a rotation of Θ by 2π, and Θ is a periodic variable, the action must be

invariant under Eq. (6.5).

Although the Lagrangian L(φ,Θ) respects a local U(1) symmetry, it really de-

scribes (redundantly) a ZN gauge theory. The point is that Θ is purely a gauge degree

of freedom; locally at least, we are free to rotate Θ to zero by means of Eq. (6.2). The

physical content of Θ therefore resides entirely in any topological obstructions that

prevent us from rotating Θ to zero globally. In particular, there are field configura-

tions such that Θ is ill–defined on a codimension–two “string” and has unit winding

number around the string. For such a configuration, Θ can be gauged away only at

the cost of making Φ N–valued in the vicinity of the string.

34

If spacetime is simply connected, then the sole purpose of Θ is to identify where

the strings are. Thus, a theory of a complex scalar field Φ with a local ZN symmetry

differs from a theory with a global ZN symmetry in that the dynamical variables

include both Φ and a ZN string degree of freedom. In principle, the action of the

theory could include a Nambu–Goto term, or a more complicated dependence on

the string worldsheet, but for the minimal version given by Eq. (6.3), the classical

string tension vanishes.?

The only effect of the string, then, is to impose a nontrivial

boundary condition on Φ — that around a loop C that links the string surface once,

Φ is not strictly periodic but is instead periodic up to the element e2πi/N of ZN

[1,4]. (Because of this boundary condition, the quantum fluctuations of Φ generate

an effective string tension.)

If the spacetime manifold M is not simply connected, then another type of ob-

struction arises that prevents Θ from being completely gauged away. If γ is a non-

contractible loop in M , then Θ may have a winding number kγ about γ,

(δΘ)γ = 2πkγ . (6.6)

This winding number kγ (modulo N) determines the boundary condition satisfied

by Φ on the loop γ; Φ is periodic up to exp[i(δΘ)γ/N ] ∈ ZN . A mod N integer k

may thus be associated with each homology cycle of M , and the k’s should also be

regarded as dynamical variables — they are to be summed over in the path integral.

Another way to describe this distinction between global and local ZN symmetry

is to note that, in the case of a local symmetry, Φ is actually to be identified with

e2πi/NΦ. Hence, the field Φ takes values not in a smooth manifold, but in an orbifold

[1,4,30] with a conical singularity at Φ = 0. That Φ takes values in C/ZN rather

than C is of no consequence, however, aside from the topological considerations noted

above.

? Of course, the effective theory that describes the low–energy limit of the Higgs phase of a U(1)gauge theory would have a positive string tension.

35

A Wilson loop operator W (C) can be expressed in terms of Θ as

W (C) = exp

[

i

N

∮

C

dx · ∂Θ

]

∈ ZN . (6.7)

This operator merely counts the (signed) number of strings that link the closed loop

C ; that is, for a given configuration of strings (and given values of the mod N integers

k associated with the homology cycles of the spacetime manifold), W (C) identifies

the boundary conditions that are satisfied by Φ on C . The flux operator F (Σ) may

also be defined as before; an insertion of F (Σ) constrains Φ to twist by e2πi/N on a

loop that links Σ once. Thus, the order parameter A(Σ, C) can be constructed as

in Eq. (4.4). This order parameter provides a criterion that specifies whether the

ZN charge is screened. If A(Σ, C) behaves as in Eq. (4.5), then ZN charges induce

quantum–mechanical effects that can be detected at long range via the Aharonov–

Bohm effect, and there is a nontrivial ZN superselection rule.

If the ZN symmetry is spontaneously broken due to the condensation of Φ, then

ZN charge is screened. But one may gain a further appreciation of the difference be-

tween global and local discrete symmetry by considering the topological defects that

result from the symmetry breakdown. When a discrete global symmetry is spon-

taneously broken, there are topologically stable domain walls. If the spontaneously

broken discrete symmetry is a local symmetry, however, then stable domain walls do

not exist [22-24]. A domain wall can end on a string, and a wall may therefore decay

by means of the spontaneous nucleation of a closed loop of string that creates a hole

in the wall (Fig. 7).

If spacetime is simply connected, then, as we have seen, the difference between

a local and global discrete symmetry rests on the existence of strings (either as dy-

namical objects or as classical sources). Even if the strings have zero tension clas-

sically, quantum fluctuations induce a string tension; this is the interpretation of κ

in Eq. (3.2). Thus, strings tend to decouple at low energy, obscuring the distinction

between local and global discrete symmetries.

36

Because of this renormalization of the string tension, the distinction between a

local and global discrete symmetry might not survive in the continuum limit of a

scalar field theory, at least if spacetime is simply connected. A continuum theory

that contains strings can be constructed only if the string tension, as well as the Φ

mass, can be chosen to be arbitrarily small in units of the cutoff Λ. Conventional

wisdom holds that this is impossible in four dimensions; dynamical strings would

induce nontrivial Φ interactions, and we could thus construct a nontrivial continuum

limit of a self–coupled scalar field theory, without introducing any gauge fields.

Finally, we conclude this section with a brief comment about discrete gauge the-

ories that contain chiral fermions. In a gauge theory with a continuous gauge group,

the fermion content is restricted by the requirement that perturbative [31] and non-

perturbative [32] gauge anomalies must cancel. It is natural to wonder whether similar

restrictions apply to a gauge theory with a discrete gauge group.

If we insist that a theory with manifest local ZN symmetry, for example, be

obtained as the low–energy limit of an underlying U(1) gauge theory, then the ZN

quantum numbers of the light fermions are restricted by the requirement that the

gauge anomalies must cancel in the underlying theory. But if the ZN gauge theory

is formulated directly, without ever introducing any gauge fields, then we claim that

there are no such restrictions on the fermion content. Regardless of the charges of the

fermions, the fermionic effective action is manifestly gauge–invariant; the only smooth

ZN gauge transformations are constant, and these have no effect on the boundary

conditions satisfied by the fermions. Nor do any cancellations occur, when we sum

over all possible boundary conditions, that render gauge–invariant Green functions

ill–defined. We also note that the nontrivial boundary conditions that distinguish a

theory with a local discrete symmetry from one with a global discrete symmetry have

no impact on the short–distance behavior of the theory. Therefore, gauge invariance

appears to have no bearing on renormalizability.?

? These conclusions about gauge anomalies are further elaborated in Ref. [33].

37

7. Nonabelian Local Discrete Symmetry

We will now extend the previous discussion to the case of a nonabelian discrete

unbroken gauge group H. Our objective is to identify the nontrivial superselection

sectors in a theory with manifest H symmetry, and hence to classify the possible

types of quantum–mechanical hair. This generalization involves some new subtleties.

In the case of an abelian gauge group, we argued that there is a distinct super-

selection sector associated with each (one–dimensional) irreducible representation of

the gauge group. It is natural to expect that this classification will continue to hold

in the nonabelian case. Indeed, we will argue that the irreducible representation of H

according to which a projectile transforms can be determined in principle by scatter-

ing the projectile off of the various cosmic strings of the H gauge theory. Equivalently,

this information can be extracted by winding loops of the various strings around the

projectile, as described in section 3.

However, in contrast to the discussion of the abelian case in section 3, we have

not succeeded in constructing a gauge–invariant operator whose eigenvalues label the

distinct superselection sectors. The strategy of seeking an operator realization of

winding a cosmic string around a region fails, in part because we are unable to find

a gauge–invariant operator that creates a cosmic string. In fact, we find that there

is a topological obstruction to implementing a global H gauge transformation in the

presence of strings (including virtual strings that can arise as quantum fluctuations).

A similar obstruction can occur, even if strings are absent, in an H gauge theory that

satisfies nontrivial boundary conditions on a multiply–connected spacetime. (See

section 8).

These mathematical statements have a physical counterpart in the remarkable

properties of the Aharonov–Bohm effect in the nonabelian case [34]. Scattering of a

projectile by a string can change the charge of the projectile. However, since a local

process cannot affect long–range hair, the irreducible representation of H according

to which projectile plus string transform must remain unchanged. Thus, a closed loop

of string must be capable of carrying H–charge. We find that this is indeed so, but

38

that, oddly, the charge “carried” by the string cannot be localized anywhere on the

string or in its vicinity.

There is no obstruction to defining a global gauge transformation that is in the

center of H; the corresponding charges are precisely those charges that cannot be

changed by Aharonov–Bohm scattering. Charge operators in the center can be con-

structed exactly as in the abelian case. But the operator realization of nonabelian

gauge charge is very elusive. We believe nonetheless that the available evidence sup-

ports the conclusion that hair in an H gauge theory is classified by the irreducible

representations of H.

We will now consider the nonabelian Aharonov–Bohm effect in greater detail.

For most of this discussion, we will find it convenient to take spacetime to be 2+1–

dimensional; rather than strings, then, the theory contains vortices, at least some of

which are stable particles. Our analysis applies to strings in 3+1 dimensions with

only minor modifications.

As we noted in section 3, a 2+1–dimensional discrete gauge theory respects a

topological superselection rule; states carry a topologically conserved vortex number.

Physically, vortex number is a type of hair because it can be detected at long range

via the Aharonov–Bohm effect, just as charge can be detected.

A vortex has hair if fields that transform nontrivially under the manifest discrete

local H symmetry are not strictly periodic on closed loops that enclose the vortex,

but are instead periodic only up to a nontrivial element Ω0 of H. (We will refer

to this boundary condition satisfied by the fields on the vortex background as the

“matching condition” imposed by the vortex.) This element Ω0 may be regarded as

the “magnetic flux” of the vortex. That is, if H is embedded in a continuous gauge

group G that has undergone the Higgs mechanism, then we may write

Ω0 = P exp

(

i

∮

C,~x

~A · d~x)

∈ H . (7.1)

Here ~A is the G gauge field, and the path–ordered integration is carried out on a

39

large oriented closed path C that encloses the vortex, beginning and ending at the

point ~x. (H is the subgroup of G that leaves invariant the symmetry–breaking order

parameter at ~x). While a finite–energy vortex configuration can be constructed with

flux Ω0 for each element Ω0 ∈ H, two distinct elements of H do not necessarily

correspond to physically distinguishable boundary conditions. This is because Ω0 is

not gauge invariant; under a gauge transformation that preserves the order parameter

at ~x, it transforms as

Ω0 → h Ω0 h−1 , h ∈ H . (7.2)

Thus, it is the conjugacy classes of H that classify the boundary conditions, and

hence the distinct types of vortex hair. (The corresponding topological statement, if

G is simply connected, is that while π1(G/H) = H classifies the closed paths in G/H

that begin and end at a specified point, the closed loops in G/H with no specified

endpoint are classified by the conjugacy classes of H [35,36].)

If the magnetic flux Ω0 of a vortex is nontrivial, then fields that transform non-

trivially under H are typically multivalued when defined on the vortex background.

As in the abelian case, this multivaluedness results in a nontrivial Aharonov–Bohm

effect. But the vortices of a theory with nonabelian local discrete H symmetry have a

characteristic feature that is not shared by the abelian case: H gauge transformations

are also multivalued on the vortex background. This multivaluedness of the discrete

gauge transformations has remarkable consequences.

These consequences are best appreciated within the context of particular exam-

ples. We will describe two examples here. In our first example, the unbroken gauge

group H is not actually a discrete group at all; instead it is a continuous nonabelian

group with two distinct connected components. We present this example in some

detail because it is quite simple and yet nicely illustrates some of our main points.

In the second example, H is a discrete nonabelian group. This example is a bit more

complicated than the first, but it is more representative of the general case.

In our first example [37,24], the gauge group is SU(2) and the order parameter Φ

is in the 5–dimensional representation of SU(2). We may express Φ as a symmetric

40

traceless 3× 3 matrix that transforms according to

Φ → Ω Φ ΩT , Ω ∈ SO(3) . (7.3)

If the expectation value of Φ in unitary gauge is

〈Φ〉 = v diag (1, 1,−2) , (7.4)

then SO(3) is broken to a subgroup that is isomorphic to O(2). The unbroken group

has two connected components. The component connected to the identity is SO(2),

containing all rotations about the z–axis. The other component contains each 180

rotation about an axis in the x-y plane.

When SO(3) is lifted to SU(2), O(2) is covered twice by a group called Pin(2).

The elements of the two connected components of Pin(2) may be parametrized as

exp

(

iθ

2σz

)

,

i σy exp

(

iθ

2σz

)

, θ ∈ [0, 4π) . (7.5)

Because H = Pin(2) is disconnected, there is a topologically stable vortex whose

“magnetic flux” is in the disconnected component of Pin(2); for a particular choice

of gauge the flux is

Ω0 = P exp

(

i

∫

~A · d~x)

= iσy . (7.6)

Thus, Ω0 does not commute with the charge operator Q = 12 σz that generates the

unbroken U(1) ⊂ Pin(2); instead we have

Ω0 Q Ω−10 = −Q . (7.7)

Therefore, a charged particle that voyages around the vortex returns to its starting

point with its charge flipped in sign. For this reason, the vortex of this model was

dubbed the “Alice vortex” in Ref. [37] — whoever circumnavigates the vortex is

reflected in the charge–conjugation looking–glass.

41

Eq. (7.7) tells us that electric charge, and hence also electric or magnetic field, are

necessarily two–valued in the background of an Alice vortex. For example, whether

two point charges have the same sign or opposite sign can be determined locally if

the charges are brought together in a region where there are no vortices nearby. (The

charges either repel or attract one another.) But whether two charges have the same

sign is not well–defined globally. If the charges are initially distantly separated, we

must bring them together to measure the relative sign of their charges. But the

outcome of the experiment depends on the path that we choose in reuniting them;

in particular, on how many times each charge winds around the vortex before they

meet. Similarly, the sign of the electric field is ambiguous because we measure the

field locally by observing the response to the field of a positive test charge, but the

sign of the test charge is not globally well–defined.

Because of the double valuedness of the electric field, there is no sensible way to

define the electric flux through a surface that contains an odd number of vortices.

In particular, then, the total electric charge of a state that contains an odd number

of vortices is ill–defined.?

Note that we are unable to take refuge in the observation

that the charge of a state can be extracted from the transformation properties of the

state under global gauge transformations, as in Eq. (2.3). The difficulty is that there

is a topological obstruction to defining a global gauge transformation on the vortex

background, precisely because the charge Q is double–valued. Since

Ω0 eiωQ Ω−1

0 = e−iωQ , (7.8)

a constant U(1) transformation on a loop enclosing a single vortex is consistent with

the matching condition imposed by the vortex only if ω = 2πn, where n is an integer.

Thus, e2πiQ, the nontrivial element of the center of H =Pin(2), is a globally–defined

? Strictly speaking, the total charge vanishes for any state of finite energy in the infinite volumelimit, because charge is (logarithmically) confined in (2+1)–dimensional QED. We ignore thistechnicality here. The reader who is concerned about this point may prefer to promote thediscussion below of charge in a two–vortex background to the case of charge in the backgroundof a string in 3+1 dimensions.

42

quantity, or the electric charge Q is well–defined modulo an integer. The globally–

defined charge, then, only tells us whether the number of fundamental charges with

|Q| = 12 is odd or even.

(This phenomenon, that a global gauge transformation can be defined on a surface

enclosing a vortex only if the transformation commutes with the matching condition

imposed by the vortex, is strongly reminiscent of an observation due to Nelson and

Manohar [38] and Balachandran et al. [38]. They found that a global gauge transfor-

mation can be implemented on a surface enclosing a magnetic monopole only if the

transformation leaves the long–range field of the monopole unchanged.)

Although the total electric charge is a meaningless notion in the background of a

single vortex, this is not so in the background of two vortices. (The two vortex sector

is topologically trivial in that the Alice vortex and antivortex are gauge equivalent.)

While the charge and electric field ~E remain globally two–valued, we can consistently

restrict ~E to one of its two branches on a loop that encloses both vortices, because

the matching conditions are trivial on this loop. In fact, we can restrict ~E to a

single branch on the whole plane at the cost of introducing a branch cut connecting

the two vortices across which ~E is required to change sign (Fig. 11). Of course,

this branch cut is a completely unphysical gauge artifact. The electric flux through

a surface that encloses both vortices is well–defined up to an overall sign, and so

the absolute value of the total charge contained in the surface can in principle be

measured. Correspondingly, a “global” gauge transformation can be implemented on

a region that contains two vortices; it is a constant transformation on the boundary of

the region, and is deformed in the interior of the region so as to vanish on the vortices

and the string connecting them. (This is analogous to the observation by Coleman

and Nelson [39], that a global gauge transformation can always be implemented on

the background on a monopole–antimonopole pair, but that the transformations that

change the long–range monopole field are required to vanish on the monopole core.)

Although the electric charge, and hence the classical “hair,” are well–defined for

a region that contains two vortices, this charge is not the same as the sum of the

43

charges of all the particles contained in the region. Since the sign of a charge cannot

be globally defined, there is no unambiguous way to add charges together.

This observation gives rise to a puzzle. As depicted in Fig. 12, we can imagine a

region that contains two vortices and two point charges, such that the total charge

vanishes as measured on a distant boundary. Suppose that the two point charges

are united, and found to be equal and opposite in sign. Call the charges ±Q. Now

let the particle with charge +Q voyage around one of the vortices, while the other

charge stays home. When the charges are reunited, both have charge −Q. Yet the

total charge, as measured at arbitrarily long–range, must not have changed. It seems

that 2Q units of charge are unaccounted for. Where did this charge go?

It is natural to suspect that the charge has been transferred to the vortex, but one

quickly sees that this is an unlikely explanation. If there does exist an excitation of

the vortex with 2Q units of charge bound to the vortex core, we expect that excitation

to be split from the vortex ground state by a finite amount of Coulomb energy.?

But

we may imagine that the charge–Q particle circles the vortex adiabatically;†

then

the charged vortex could not get excited. Moreover, there is no indication within

the semiclassical approximation that the vortex has charged excitations. Charged

excitations of solitons arise semiclassically from quantizing the global charge rotor

degree of freedom of the soliton. But we have seen that global charge rotations of the

Alice vortex do not exist.

Even more telling, we may imagine that the charged particle circumnavigates the

vortex while distantly separated from it. Since all charged fields have finite mass,

there seems to be no mechanism by which the particle could transmit charge to the

vortex at arbitrarily long range.

? More correctly, since the Coulomb energy of an isolated charged particle is infinite in twodimensions, we should compare two configurations with vanishing total charge. The configu-ration with nonzero charge localized on the vortex should be split from the configuration withzero charge on the vortex by a finite gap.

† We need to do some work to move the charged particle in the fields of the other particles, butthis work may be made as small as desired.

44

Even if charge cannot be carried by an isolated vortex, conservation of the electric

flux at spatial infinity requires that charge can be carried by a vortex pair. We seem,

though, to require a miracle, for the charged particle must be able to excite a charged

state of the vortex pair without actually transferring any charge to the pair. It is the

two–valuedness of the electric field that makes it possible to perform this miracle.

Figure 13 shows one branch of the static two–valued electric field of a point charge

Q in the vicinity of a vortex pair, for a sequence of positions of the point charge. The

total electric charge as measured at spatial infinity (on this branch) is also assumed to

be Q. When the charge reaches the cut that connects that two vortices, it disappears

behind the cut just as its image charge −Q emerges from behind the cut. The electric

flux −2πQ emanating from the image charge returns to the second sheet through the

cut, and the flux 2πQ emanating from the original charge likewise returns to the first

sheet through the cut. After the charge has passed the cut, then, an observer on

a closed surface that encloses the two vortices, but not the point charge, measures

electric flux 4πQ through the surface, and infers that 2Q units of charge are inside.

In fact, though, this charge is not localized anywhere. The electric flux through any

closed box on the two–sheeted surface vanishes, if the box does not contain the point

charge, its image, or either vortex. Nevertheless, the vortex pair, initially in its zero–

charge ground state, has been excited to a charge–2Q state after the point charge has

passed through. The vortex pair now has hair. The electric field lines trapped by the

pair cause the vortices to repel each other.‡

On a background with an even number of vortices at specified positions, the

classical electric field on a single sheet is not uniquely determined by the positions

and values of all point charges on that sheet. One must also know the charge “carried”

by each cut. As the point charges move with respect to this background, the values of

the point charges may change sign, but the “charge” on the cuts also changes, in such

a way that the total charge, the “hair,” stays constant. If |Q| = 12 is the quantum of

‡ That a pair of Alice vortices or a loop of Alice string can carry unlocalized electric charge hasalso been noted by Alford et al. [44]. They propose the apt name “Cheshire charge” for thisphenomenon.

45

charge, then the sum of the values of the point charges can change by only an integer.

This is in accord with our earlier observation that e2πiQ is a well–defined quantum

number, even when the number of vortices is odd.

(A monopole–antimonopole pair, where the monopole has a nonabelian magnetic

field, also has charged excitations [39]. In that case, however, there are light charged

fields, the nonabelian gauge fields, that carry the charge of the excitation. The

charged excitations of a pair of Alice vortices are quite different, as we have just

seen.)

The above discussion of a pair of Alice vortices in two spatial dimensions gen-

eralizes immediately to a loop of Alice string in three spatial dimensions. From the

nonabelian Aharonov–Bohm effect, we infer that a loop of Alice string can have clas-

sical hair. A loop can carry any integer amount of electric charge, and a loop with

charge Q and length L has a Coulomb energy of order Q2e2/L. A sufficiently large

charged loop is therefore metastable; all charged particles are massive, so the loop

cannot reduce its charge by emitting charged particles.

The oscillations of the loop cause it to emit both gravitational and electromagnetic

radiation. In order of magnitude, the gravitational and electromagnetic power are

Pgrav ∼ GM2/L2 , Pem ∼ Q2e2/L2 , (7.9)

where M is the mass of the loop. The loop loses energy and shrinks. It may eventually

reach a stable configuration in which the string tension is balanced by the electric

flux trapped by the loop. However, the mass of this configuration is of order

M ∼ Q e√µ , (7.10)

where µ is the string tension, and so is comparable to the mass of Q charged vector

bosons. Thus, the decay of this minimal charged loop to vector bosons may be

kinematically allowed.

46

Having unraveled the interplay of the nonabelian Aharonov–Bohm effect and

classical hair in the case of the Alice vortex, we are now prepared to return to our

main interest — quantum–mechanical hair. Again, it is useful to consider a particular

example in some detail.

Our example [36,40,41] will be the same model as before, but with a different

pattern of symmetry breakdown. We now suppose that the order parameter Φ has,

in unitary gauge, the expectation value

〈Φ〉 = v diag (1 + δ, 1 − δ,−2) ; (7.11)

this reduces to Eq. (7.4) in the limit δ → 0. For δ 6= 0, the unbroken subgroup of

SO(3) is the four–element subgroup D2 ∼ Z2 × Z2, which contains the identity and

180 rotations about each of the x, y and z axes. When SO(3) is lifted to SU(2), D2

is covered twice by the eight–element quaternionic group

Q = ±1,±iσx,±iσy,±σz . (7.12)

Thus, H = Q is a discrete nonabelian group.

The model with unbroken Q symmetry, unlike the model with unbroken Pin(2)

symmetry, has no massless gauge fields. It also differs from the Pin(2) model in

another significant respect that concerns the properties of vortices. The conjugacy

classes of Q are

1, −1, ±iσx, ±iσy, ±iσz . (7.13)

As we have noted, the conjugacy classes of a discrete gauge group H classify the

values of the “magnetic flux” or topological charge. Hence, the Q model contains

four distinct types of vortices.

Multiplication of conjugacy classes is in general ambiguous, and there is a cor-

responding ambiguity when two vortices are patched together to make a vortex pair

[35,36]. For example, the product of two elements of the conjugacy class ±iσx

47

can be either 1 or −1, which are two distinct classes. When two vortices that both

represent the class ±iσx are brought together, then, they may or may not be able

to annihilate each other, depending on how the patching has been performed. This

phenomenon is actually closely analogous to the property we observed in the Pin(2)

model, that whether two charged particles have equal or opposite charges is depen-

dent on the choice of a path that connects the two particles. In general, if a vortex

with magnetic flux Ω1 is transported around a vortex with magnetic flux Ω2, its flux

becomes conjugated,

Ω1 → Ω2 Ω1 Ω−12 . (7.14)

So if a topologically trivial pair of vortices is produced, say with flux iσx and −iσx,

and one of the vortices voyages around a ±iσy vortex, then the magnetic flux of

the pair has become −1 when the vortices are reunited (Fig. 14). This example

illustrates why magnetic flux and topological charge can be globally defined only

up to conjugacy. (The corresponding phenomenon in three spatial dimensions is

that ±iσx and ±iσy strings cannot pass through each other without becoming

entangled [36,40].) Since topological quantum numbers do not prevent the −1vortex from decaying to a pair of ±iσa vortices, this vortex may be unstable, at

least for a range of values of the parameters of the model.

The number of conjugacy classes of a finite group H is also the num-

ber of inequivalent irreducible representations of H. The irreducible repre-

sentations of Q include, aside from the defining two–dimensional representa-

tion and the trivial one–dimensional representation, three other nontrivial one–

dimensional representations. The one–dimensional representations all repre-

sent ±1 by the identity and represent the other elements of Q as follows:

48

Representation ±iσx ±iσy ±iσz10 1 1 1

1x 1 −1 −1

1y −1 1 −1

1z −1 −1 1

Table 1

The five irreducible representations are the five distinct values that the “charge” can

assume in the Q model.

The charge of a state in an H gauge theory is ill–defined unless the magnetic

flux of the state has a value in the center of H. For Q, the center is 1,−1. In

particular, then, Q–charge is ill–defined in the background of a single ±iσx vortex

but well–defined in the background of a pair of ±iσx vortices. This charge, as well

as the vortex number, is a type of quantum–mechanical hair that can be detected at

long–range by means of the Aharonov–Bohm effect.

We say that Aharonov–Bohm scattering of a charged projectile by one member

of a vortex pair is nonabelian if charge is transferred to the pair and is abelian if no

charge transfer occurs. As for the case of the Alice vortex, we can see that nonabelian

Aharonov–Bohm scattering must be possible in the Q–model. Before turning to the

nonabelian effect, however, let us first note that the abelian Aharonov–Bohm effect

can in principle be used to unambiguously determine the charge of a projectile in the

Q–model.

First, charges in the center of a discrete gauge group H can always be detected

by vortices with magnetic flux in the center of H; the Aharonov–Bohm scattering

involving these vortices is always abelian. In the case of Q, the vortex with flux

Ω0 = −1 distinguishes the doublet representation, with which is has a nontrivial

Aharonov–Bohm effect, from the four singlet representations, with which it has no

Aharonov–Bohm effect.

49

Aharonov–Bohm scattering of the singlet charges is necessarily abelian. We may

distinguish among the four distinct one–dimensional representations by scattering off

the three remaining types of vortices. We can read off from Table 1 that the ±iσxvortex, for example, scatters projectiles with charge 1y or 1z , but does not scatter

projectiles with charge 10 or 1x, while the ±iσy vortex scatters 1x and 1z , but not

10 or 1y. So scattering off these two vortices evidently distinguishes the four singlet

representations of Q.

We can also invoke the abelian Aharonov–Bohm effect to measure the charge in

a large region, if the magnetic flux in the region is in the center of Q. As described

in section 3, the charge can be detected as the Aharonov–Bohm phase acquired by a

vortex that circumnavigates the region.

The nonabelian Aharonov–Bohm effect occurs in the Q model when a projectile

in the doublet representation of Q scatters off a vortex whose magnetic flux is not in

the center of Q. Consider, for example, a pair of vortices, each with magnetic flux

±iσa. If the pair initially has the trivial 10 charge, and a projectile in the doublet

representation scatters off one member of the pair, then the pair becomes excited to

the 1a representation. The charge carried by a vortex pair is restricted to the singlet

representations of Q, as the center charge of Q must be well–defined even in the

background of a single vortex.

We can gain some insight into the mechanism of nonabelian Aharonov–Bohm

scattering in the Q–model by contemplating a model with a symmetry–breaking

hierarchy. If δ in Eq. (7.11) satisfies δ << 1, then the model has three different

energy regimes. At short distances SU(2) is effectively restored; at intermediate

distances, Pin(2) is a good symmetry; and at long distances only the Q symmetry

survives. Then our earlier description of a charged particle interacting with an Alice

vortex applies to Aharonov–Bohm scattering at intermediate energy of a doublet off

a pair of ±iσy vortices with intermediate separation. A projectile in the doublet

representation of Q must be in a |Q| = n + 12 representation of Pin(2), where n is

a non–negative integer, and excites an initially uncharged state of the vortex pair to

50

a state with charge |Q| = 2n + 1. A pair of ±iσy vortices in the |Q| = 2n + 1

representation of Pin(2) transforms as the 1y representation of Q. At long distances,

the Pin(2) charge of the vortex pair becomes screened by the condensate, but the

Q charge is a type of hair that cannot be screened. Nor can the Q charge become

screened as the separation of the vortex pair smoothly varies from an intermediate

distance to a long distance.

We see, now, that the Q–model lends support to our general claims. Quantum–

mechanical hair in the model is classified by magnetic flux, and in the sectors with

magnetic flux in the center of Q, by an irreducible representation of Q. Furthermore,

Q charge can be carried by an isolated vortex pair.

In general, if H is a discrete gauge group, then a vortex with magnetic flux in

the conjugacy class h0 can be patched together with a vortex with flux in the class

h−10 to make a configuration with trivial flux. The possible values of H charge that

this pair can carry are determined as follows: H can act on a representative h0 of a

conjugacy class according to

h: h0 → h h0 h−1 , h ∈ H . (7.15)

The representatives of a class thus transform as a (in general reducible) representation

of H with dimension equal to the order of the class. The irreducible representations of

H contained in this representation are the allowed values of the charge of the vortex

pair.

For example, the class ±iσy of Q transforms as the representation 10 + 1y of

Q. The class

P (θ) = iσy exp(

iθ

2σz

)

(7.16)

of Pin(2) transforms as

exp(

iω

2σz

)

:P (θ) → P (θ + 2ω) ; (7.17)

this is an infinite–dimensional representation of Pin(2) that contains all the integer–

|Q| irreducible representations.

51

A gauge–invariant operator can be constructed corresponding to each element in

the center H of the unbroken gauge group H. Each such operator may be defined

as in Eq. (3.3), and an order parameter A(Σ, C) can be formulated as in Eq. (4.4)

that probes how the local H symmetry is realized. However, these statements require

a qualification. If H is embedded in a continuous gauge group G that undergoes

the Higgs phenomenon, then the operator F (Σ) corresponding to the H charges is

invariant under local G transformations only if H is contained in the center of G. If

this is not the case, then the H charges must be constructed in a low–energy effective

theory with local H symmetry, as we described in section 6.

If the unbroken discrete gauge group H is nonabelian, though, we have argued

that there is a charge superselection sector associated with each irreducible repre-

sentation of H. This is a richer classification than can be probed by the H charge

operators alone. But we have been unable to construct gauge–invariant operators

whose eigenvalues distinguish the various sectors. Our difficulties result from the

peculiar properties of the nonabelian Aharonov–Bohm effect.

In particular, in two spatial dimensions, the H charge contained in a region is in

general ill–defined, unless the “magnetic flux” contained in the region is in the center

H of H. Still, one can hope that an H charge superselection rule can be formulated

in the sector of the theory with magnetic flux in H . Similarly, in three dimensions,

H charge should be well–defined in a region that contains cosmic strings as long as

no “branch cuts” intersect the boundary of the region.

The task of assigning a definiteH charge to a bounded region is complicated, how-

ever, by quantum fluctuations. Virtual vortex pairs near the boundary of the region

cause the enclosed magnetic flux to fluctuate. Therefore, computing the expectation

value of the charge necessarily involves averaging over values of the enclosed flux for

which the charge is ill–defined. Because this is merely a surface effect, though, we

expect that a well–defined H charge can be extracted in the limit of infinite volume.

A related problem is that a global H gauge transformation cannot be defined when

vortices are present. This problem is even more acute for a discrete gauge group like

52

Q than for a continuous nonabelian group like Pin(2). In the case of Pin(2), a global

U(1) gauge transformation can be defined acting on a state that contains an even

number of vortices. This transformation is nontrivial at spatial infinity, but can be

smoothly deformed to the identity on all vortices and branch cuts. If H is discrete, no

such smooth deformation is possible. Only the gauge transformations in the center

H of H can be performed on a general background, for the other transformations fail

to preserve the nontrivial matching conditions on the branch cuts.

A mathematical purist might conclude, then, that only the H charges can be

defined in the presence of vortices; even virtual vortex pairs deep inside a region

obstruct the global H transformations acting on the region. We are reluctant to

accept so drastic a conclusion. As we have argued, the Aharonov–Bohm phases

acquired by various vortices or cosmic strings that wind around a region provide us

in principle with sufficient information to determine the H charge contained in the

region.

We have been unable to find an operator realization of this experiment, because

we do not know how to construct any gauge–invariant operator that creates a cosmic

string or vortex pair, if the string or vortex exhibit the nonabelian Aharonov–Bohm

effect. This in itself is rather curious, especially in the case of the vortex, which can

be a stable particle in a (2 + 1)–dimensional field theory. There should be S–matrix

elements with vortices in the asymptotic in and out states, but we do not know how

to obtain these S–matrix elements by applying the LSZ procedure to gauge–invariant

Green functions.

We note in passing that a vortex bound to a charge particle provides a realization

of nonabelian statistics in 2 + 1 dimensions, if the vortex and particle exhibit the

nonabelian Aharonov–Bohm effect. A pair of gauge–equivalent “identical” particles

undergoes a nonabelian transformation when the particles are interchanged. The

physical interpretation of this construction is rather obscure, since an isolated particle

has ill–defined charge, and a pair of particles can carry unlocalized charge.

53

8. The Charge of a Closed Universe

An important distinction between gauge and global symmetries is emphasized

in Ref. [1]. Because a gauge symmetry arises when the variables used to describe

a system are redundant, gauge symmetries, unlike global symmetries, are intrinsi-

cally exact. A gauge symmetry that is weakly broken by a small perturbation is an

oxymoron.

This distinction has a familiar consequence in connection with the topology–

changing interactions that may occur in quantum gravity [5]. Figure 15 depicts a

wormhole in Euclidean spacetime, a process in which a parent universe gives birth

to a tiny closed baby universe that is disconnected from the parent, and the baby

universe carries away a unit of a globally conserved charge. An observer in the parent

universe sees the charge disappear, and interprets this event as a violation of the

global conservation law.

The corresponding process in which a nonvanishing amount of a locally conserved

charge is swallowed by a wormhole must be forbidden. An observer in the parent

universe would interpret this event as a violation of a local conservation law, which

we know to be impossible. It is familiar how wormhole physics is reconciled with gauge

invariance in the case of a locally conserved charge, like electric charge, that couples

to a massless gauge field. The baby closed universe is required to carry vanishing

total charge. This requirement is a consequence of the Gauss law — charge can be

expressed as a surface integral, but the baby universe has no boundary. From the

perspective of the parent universe, a region contained in the universe can pinch off

and become a disconnected baby universe only if the region has no hair that can be

detected far outside the region.

This heuristic understanding of why wormholes respect gauge symmetries extends

to the case of local discrete symmetries; discrete gauge charges, although screened

classically, have quantum mechanical hair that can be detected at the boundary of a

manifold by means of the Aharonov–Bohm effect. In a manifold without boundary,

such hair is absent, and so the total gauge charge must be trivial.

54

In the case of an abelian discrete gauge symmetry, the charge operator defined in

section 3 explicitly expresses the charge enclosed by a surface in terms of fields on that

surface. Eq. (3.1) is the analog of the Gauss law for a discrete charge, and suffices to

ensure that a closed manifold contains zero total charge. That is, a closed manifold

can be obtained from a manifold with boundary Σ in the limit in which Σ shrinks to

a point; in this limit F (Σ) approaches one, and the charge enclosed by Σ vanishes. In

the nonabelian case, although we are unable to construct a gauge–invariant operator

realization of the Gauss law, we have seen that the hair can in principle be detected

by cosmic strings at the boundary.

In the nonabelian case, however, the “total charge” is not necessarily the same

as the sum of all the point charges contained in the universe. If the universe is not

a simply connected manifold, and the gauge group is nonabelian, then addition of

charges may be ambiguous, just as we found in the background of a string or vortex

pair. To illustrate this phenomenon, consider the model with unbroken gauge group

H = Pin(2) that we described in section 7. A universe with a handle attached to

it might be an “Alice universe” on which the electric field ~E and electric charge Q

are double–valued; if we restrict ~E and Q to a single branch, then, there must be a

branch cut on a closed surface contained in the handle, where ~E and Q change sign.

In such an Alice universe, a charged particle that voyages through the handle returns

to its starting point with its charge flipped in sign [42].

Of course, this process in which a charge changes sign by traversing a handle

must not modify the electric field as measured far away from the handle. Indeed,

just as we saw that a charged particle that passes through a loop of Alice string

must transfer charge to the loop, a charged particle that traverses an “Alice handle”

transfers charge to the handle. The charge of the handle can be defined by means of

the electric flux through a surface that is the boundary of a region that contains the

handle, as long as this surface does not intersect the branch cut. While the handle

itself has a well–defined electric charge, this charge is not localized anywhere inside

the handle. The branch cut appears to be the source of the flux, but the cut is an

unphysical gauge artifact, and no actual charge resides there.

55

In general, if ~E is restricted to a single branch, then the “total charge” is the sum

of the point charges and of the “charges” on all of the branch cuts; this is the quantity

that must vanish in a closed universe. Figure 16 depicts the electric field of a single

charged particle in an Alice universe; the charge Q of the particle is compensated by

the charge −Q of the handle.

Since the element e2πiQ is in the center of Pin(2), it commutes with the nontrivial

transition function on the branch cut, and a handle is therefore not permitted to

carry this charge. This means that Q modulo an integer is a well–defined additive

quantum number that can be computed by summing the values of the point charges.

In particular, then, the number of elementary point charges with |Q| = 12 must be

even in a closed universe.

This discussion of charge in the Pin(2) model is also applicable to the case of

nonabelian discrete gauge symmetry, as we argued in section 7.

9. Conclusions

In a gauge theory, a superselection rule arises whenever there are states that are

endowed with properties that can be detected at arbitrarily long range. In this sense,

the superselection rules provide a classification of the types of “hair” that a localized

object can carry. This paper has aimed to clarify the nature of the superselection

rules in theories with local symmetry. In particular, we have examined the notion of

“quantum–mechanical hair” that is invisible classically, but can be detected via the

Aharonov–Bohm effect. Such quantum–mechanical hair can in principle be carried

by a black hole.

In the case of quantum–mechanical hair that is associated with an abelian discrete

gauge symmetry, our discussion has been reasonably complete. We constructed a

charge operator whose eigenvalues distinguish the various superselection sectors of

the theory, and formulated a nonlocal order parameter that probes the realization of

the local symmetry. Our construction translates into operator language the heuristic

idea that the charge inside a region can be measured as the Aharonov–Bohm phase

56

that is acquired by a cosmic string that winds around the region. The charge operator

provides, in particular, an operator description of the quantum–mechanical hair on a

black hole.

In the case of a nonabelian discrete gauge symmetry, our grasp of the superselec-

tion rules is in a less satisfactory state. We proposed that, as in the abelian case, the

charge superselection sectors are classified by the irreducible representations of the

gauge group, and we described how the representation content can be inferred from

measurable Aharonov–Bohm phases. However, we succeeded in constructing gauge–

invariant charge operators only for the charges in the center of the gauge group. We

could not translate the procedure for measuring Aharonov–Bohm phases into oper-

ator language because, curiously, we could not find a gauge–invariant operator that

creates the desired cosmic string. This difficulty is closely related to a remarkable

feature of the nonabelian Aharonov–Bohm effect — a loop of cosmic string can carry

charge, even though the charge cannot be localized anywhere on the string or in its

vicinity.

The implications of the Aharonov–Bohm effect, and of local discrete symmetry,

are surprisingly deep; we feel that these implications have not yet been fully explored.

We expect, in particular, that further investigation of the nonabelian Aharonov–Bohm

effect will prove to be highly rewarding.

Alford, March–Russell, and Wilczek have independently investigated the prop-

erties of quantum–mechanical hair and the nonabelian Aharonov–Bohm effect in

Ref. [43]. They, together with Benson and Coleman are pursuing further consequences

of the nonabelian Aharonov–Bohm effect [44].

57

Acknowledgements

J.P. gratefully acknowledges a helpful discussion with Sidney Coleman concerning

the Alice string and Cheshire charge. L.K. acknowledges useful discussions with Frank

Wilczek about discrete gauge charges and local discrete symmetry. We have also

benefited from conversations with Frank Accetta, Jim Hughes, Joe Polchinski, David

Politzer, Soo–Jong Rey, Opher Shapira, and Lenny Susskind.

58

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61

Figures

Fig. 1. An insertion of the flux operator F (Σ) introduces a ZN discontinuity on a

hypersurface Ω whose boundary is Σ. The gauge field has a ZN winding number

on a closed loop C that links Σ.

Fig. 2. Σ may be regarded as the worldsheet of a loop of cosmic string.

Fig. 3. A cosmic string that adiabatically winds around Σ can detect the charge inside.

Fig. 4. Spacetime view of the detection procedure. A charge is detected if its worldline

crosses the hypersurface Ω that is bounded by Σ.

Fig. 5. The ‘t Hooft operator B(C, Σ) becomes the flux operator F (Σ) as the loop C

shrinks to a point.

Fig. 6. A sequence of closed loops, all containing the common point P , that covers Σ.

Fig. 7. Spontaneous nucleation and subsequent expansion of a loop of string causes a

domain wall to decay.

Fig. 8. A closed loop C and closed surface Σ with linking number 1.

Fig. 9. Phase diagram of the Z2 gauge-spin system.

Fig. 10. The set of plaquettes (shaded) dual to a set Σ of links (bold) of the dual lattice,

in three Euclidean dimensions. Dotted lines are links of the dual lattice, and

solid lines are links of the original lattice. In four Euclidean dimensions, the

bold line should be interpreted as a slice through the surface Σ.

Fig. 11. Two Alice vortices connected by a branch cut. The electric field is single-valued

and smooth on Σ.

Fig. 12. A charged particle voyages around a vortex and returns with the sign of its

charge changed. 2Q units of charge have been deposited inside Σ.

Fig. 13. One branch of the static electric field of a point charge in the vicinity of a vortex

pair, for a sequence of positions of the point charge.

Fig. 14. When a vortex with flux Ω1 is transported around a vortex with flux Ω2, its

flux becomes conjugated.

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Fig. 15. A globally conserved charge disappears down a wormhole.

Fig. 16. One branch of the electric field of a point charge in an Alice universe.

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